Derivation of the Langevin Equation from the Microcanonical Ensemble

Abstract

When writing down a Langevin equation for the time evolution of a “system” in contact with a thermal bath, one typically makes the implicit (and often tacit) assumption that the thermal environment is in equilibrium at all times. Here, we take this assumption as a starting point to formulate the problem of a system evolving in contact with a thermal bath from the perspective of the bath, which, since it is in equilibrium, can be described by the microcanonical ensemble. We show that the microcanonical ensemble of the bath, together with the Hamiltonian equations of motion for all the constituents of the bath and system together, give rise to a Langevin equation for the system evolution alone. The friction coefficient turns out to be given in terms of auto-correlation functions of the interaction forces between the bath particles and the system, and the Einstein relation is recovered. Moreover, the connection to the Fokker–Planck equation is established.

1. Introduction

The Langevin equation [1,2,3,4,5,6] is a well-established and extremely successful model for describing how a system evolves under the combined influence of deterministic and rapidly fluctuating (“random”) forces. Typically, the random forces and their stochastic nature are the result of many (microscopic) degrees of freedom, which change extremely quickly compared to the “slow” degrees of freedom of the system. A paradigmatic application is Brownian motion [2], i.e., the diffusive motion of a small particle in a fluid or gas, as schematically illustrated in Figure 1 (the Langevin equation for this setup is explicitly stated in (55)). Generally, a macroscopic thermodynamic system plays the role of a “thermal bath” or “environment” for a small system of interest. A common standard assumption when investigating the effective stochastic dynamics of such systems is that the thermal bath remains in equilibrium at all times [6].

This equilibrium assumption is justified by an extreme separation of time scales. The bath possesses a fast intrinsic time scale $\tau$ on which perturbations away from equilibrium decay rapidly (typically, $\tau$ is related to the molecular collision time, which is about ${10}^{-13}\mathrm{s}$ in water). The dynamics of the system occur on time scales much slower than this characteristic time scale of the bath. For instance, for a colloidal particle, the fastest time scale is the velocity relaxation time due to viscous friction, which is several orders of magnitude larger than $\tau$ (it is about ${10}^{-8}\mathrm{s}$ for a colloidal sphere of radius $0.1\mathsf{\mu }\mathrm{m}$ in water). Its (diffusive) spatial motion occurs on even slower time scales (about $\mathrm{m}\mathrm{s}$). Moreover, in typical experiments, external time-dependent variations in forces, etc., exerted on the system do not directly interfere with the bath and vary at times much larger than $\tau$. Note that we completely disregard that perturbations might create slow collective “degrees of freedom” in the bath. Even though this is known to be the case in dense environments, the modifications of the system dynamics due to such slow collective modes in the bath seem to be of little practical relevance for typical systems usually modeled by Langevin equations. Put differently, the theory that we are going to develop here is a proper description for systems for which such slow bath modes do not have observable consequences.

Hence, at any instance of slow “system time”, i.e., at any time resolution $\Delta t$ of interest in typical experiments that capture the system behavior (with $\Delta t\gg \tau$ being microscopically large but macroscopically small), the bath is in an equilibrium state with a well-defined energy. Energy changes occur only via processes connected to changes in the system state, e.g., via a displacement of the colloidal particle. For the fast microscopic degrees of freedom of the bath. such processes, occurring over $\Delta t$ time scales, are quasi-statically slow transformations from a given equilibrium state to a new one. Since the bath is isolated from the rest of the world, each of these equilibrium states can be characterized by the microcanonical ensemble for the current value of bath energy.

Based on a classical Hamiltonian description of the system and bath together (Section 2), and the representation of the bath state by the microcanonical ensemble, we analyze the system dynamics on $\Delta t$ time scales. We show that individual $\Delta t$ steps are statistically independent (Section 3.1), implying that the system dynamics are Markovian on $\Delta t$ time scales. For individual time steps $\Delta t$, we calculate averages and correlations of changes in the system state (Section 3.2). In linear order $\Delta t$, these moments turn out to be identical to those generated by a standard Langevin equation for the system dynamics alone (Section 3.3). Hence, we demonstrate that the standard Langevin equation represents an “effective” evolution equation for a system in contact with a bath, both together described on the microscopic level by a classical Hamiltonian. We also derive from the Hamiltonian description that the proper rules of stochastic calculus [3,4] for the effective Langevin equation are given by the Stratonovich interpretation (Section 3.4).

2. Model

We consider the setup schematically illustrated in Figure 1. A macroscopically large thermodynamic system plays the role of a thermal bath or environment for a (small) system of interest. We denote all parameters and coordinates, etc., related to the bath by small letters, and those specifying the system by capital letters; Greek indices denote particle numbers and Latin indices denote components of three-dimensional (coordinate) vectors. We collect all bath degrees of freedom in the super-vector $\varphi =(p,q)=({\mathit{p}}_1,{\mathit{p}}_2,\dots ,{\mathit{p}}_n,{\mathit{q}}_1,{\mathit{q}}_2,\dots ,{\mathit{q}}_n)$ and all system degrees of freedom in $\mathrm{\Phi }=(P,Q)$$=({\mathit{P}}_1,{\mathit{P}}_2,\dots ,{\mathit{P}}_N,{\mathit{Q}}_1,{\mathit{Q}}_2,\dots ,{\mathit{Q}}_N)$. We assume that $N⋘n$. The total Hamiltonian of the bath and system together is

$$H(\varphi ,\mathrm{\Phi })=\underset{=H_{\mathrm{bath}}(\varphi ,Q)}{\underbrace{H_{\mathrm{bulk}}\left(\varphi \right)+H_{\mathrm{int}}(q,Q)}}+H_{\mathrm{sys}}(\mathrm{\Phi },\lambda ),$$

(1a)

with

$$\begin{array}{cc}\hfill H_{\mathrm{bulk}}\left(\varphi \right)& =\sum _{\mu }\frac{{\mathit{p}}_{\mu }^2}{2m}+\sum _{\mu <\nu }{U}_{\mathrm{bulk}}\left(\right|{\mathit{q}}_{\mu }-{\mathit{q}}_{\nu }\left|\right),\hfill \end{array}$$

(1b)

$$\begin{array}{cc}\hfill H_{\mathrm{int}}(q,Q)& =\sum _{\mu ,\nu }{U}_{\mathrm{int}}\left(\right|{\mathit{q}}_{\mu }-{\mathit{Q}}_{\nu }\left|\right),\hfill \end{array}$$

(1c)

$$\begin{array}{cc}\hfill H_{\mathrm{sys}}(\mathrm{\Phi },\lambda )& =\sum _{\mu }\frac{{\mathit{P}}_{\mu }^2}{2M}+U(Q,\lambda ).\hfill \end{array}$$

(1d)

The interactions between the bath particles in the bulk are captured in $U_{\mathrm{bulk}}\left(\right|{\mathit{q}}_{\mu }-{\mathit{q}}_{\nu }\left|\right)$, and the interaction between bath particles and the system is captured in $U_{\mathrm{int}}\left(\right|{\mathit{q}}_{\mu }-{\mathit{Q}}_{\nu }\left|\right)$. Both depend on particle–particle distances. We combine $H_{\mathrm{bulk}}\left(\varphi \right)$ and $H_{\mathrm{int}}(\varphi ,Q)$ into the bath Hamiltonian $H_{\mathrm{bath}}(\varphi ,Q)$ because, for the effective system dynamics that we are aiming at, the system energy should be determined by $H_{\mathrm{sys}}(\mathrm{\Phi },\lambda )$ alone without reference to bath degrees of freedom. The potential $U(Q,\lambda )$ represents the potential forces applied to the system by external means; they affect only positional degrees of freedom (not momenta). The potential is, moreover, allowed to depend on time via the protocol $\lambda =\lambda \left(t\right)$, varying noticeably only on time scales much larger than $\tau$.

2.1. Dynamics

The Hamilton equations of motion are [7]

$$\begin{array}{cc}\hfill {\dot{\mathit{p}}}_{\mu }& =-\frac{\partial H}{\partial {\mathit{q}}_{\mu }}=-\frac{\partial H_{\mathrm{bath}}}{\partial {\mathit{q}}_{\mu }},\hfill \end{array}$$

(2a)

$$\begin{array}{cc}\hfill {\dot{\mathit{q}}}_{\mu }& =\frac{\partial H}{\partial {\mathit{p}}_{\mu }}=\frac{{\mathit{p}}_{\mu }}{m},\hfill \end{array}$$

(2b)

$$\begin{array}{cc}\hfill {\dot{\mathit{P}}}_{\mu }& =-\frac{\partial H}{\partial {\mathit{Q}}_{\mu }}=-\frac{\partial H_{\mathrm{int}}}{\partial {\mathit{Q}}_{\mu }}-\frac{\partial H_{\mathrm{sys}}}{\partial {\mathit{Q}}_{\mu }},\hfill \end{array}$$

(2c)

$$\begin{array}{cc}\hfill {\dot{\mathit{Q}}}_{\mu }& =\frac{\partial H}{\partial {\mathit{P}}_{\mu }}=\frac{{\mathit{P}}_{\mu }}{M},\hfill \end{array}$$

(2d)

where the partial derivative symbols with respect to a vector quantity denote the corresponding gradient vector. Following standard convention in physics, we use the same symbols for the solutions of these equations as for the coordinates in phase space $(\varphi ,\mathrm{\Phi })$, just endowing them with a time argument, which we write here as an index; if useful, we will also explicitly state the dependence on the initial values. The formal solutions for the various components $p=({\mathit{p}}_1,{\mathit{p}}_2,\dots ,{\mathit{p}}_n)$, $q=({\mathit{q}}_1,{\mathit{q}}_2,\dots ,{\mathit{q}}_n)$ and $P=({\mathit{P}}_1,{\mathit{P}}_2,\dots ,{\mathit{P}}_N)$, $Q=({\mathit{Q}}_1,{\mathit{Q}}_2,\dots ,{\mathit{Q}}_N)$ are

$$\begin{array}{cc}\hfill p_t({\varphi }_0,{\mathrm{\Phi }}_0)& =p_0-{\int }_{t_0}^t\mathrm{d}{t}^{\prime }\frac{\partial H_{\mathrm{bath}}({\varphi }_{t^{\prime }},Q_{t^{\prime }})}{\partial q},\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill q_t({\varphi }_0,{\mathrm{\Phi }}_0)& =q_0+{\int }_{t_0}^t\mathrm{d}{t}^{\prime }\frac{p_{t^{\prime }}}{m},\hfill \end{array}$$

(3b)

$$\begin{array}{cc}\hfill P_t({\varphi }_0,{\mathrm{\Phi }}_0)& =P_0-{\int }_{t_0}^t\mathrm{d}{t}^{\prime }\left[\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q}+\frac{\partial H_{\mathrm{sys}}({\mathrm{\Phi }}_{t^{\prime }},{\lambda }_{t^{\prime }})}{\partial Q}\right],\hfill \end{array}$$

(3c)

$$\begin{array}{cc}\hfill Q_t({\varphi }_0,{\mathrm{\Phi }}_0)& =Q_0+{\int }_{t_0}^t\mathrm{d}{t}^{\prime }\frac{P_{t^{\prime }}}{M},\hfill \end{array}$$

(3d)

for an initial configuration $({\varphi }_0,{\mathrm{\Phi }}_0)$ at time $t_0$. Here, the notation $\frac{\partial H_{\mathrm{bath}}({\varphi }_{t^{\prime }},Q_{t^{\prime }})}{\partial q}$ means that we evaluate the derivative of $H_{\mathrm{bath}}(\varphi ,Q)$ with respect to q (this is the $3n$-dimensional gradient vector) and evaluate it along the trajectory $({\varphi }_{t^{\prime }},Q_{t^{\prime }})$, and likewise for similar expressions.

According to the time scale separation discussed above, $(q_t,p_t)$ vary extremely rapidly on the molecular collision time scale $\tau$, while $(Q_t,P_t)$ are virtually constant on that scale but change quickly on macroscopic time scales. The system degrees of freedom $(Q_t,P_t)$ therefore evolve on a mesoscopic time scale $\Delta t$ with $\tau \ll \Delta t\ll (\mathrm{macroscopic} \mathrm{time} \mathrm{scale})$. For the time evolution over such a time interval $\Delta t$, we can write

$$\begin{array}{cc}\hfill Q_{\Delta t}(\varphi ,\mathrm{\Phi })& =Q+{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\frac{P_{t^{\prime }}}{M}\hfill \\ \hfill & =Q+\frac{P}{M}\Delta t+\mathcal{O}(\Delta t^2)=Q+\mathcal{O}(\Delta t),\hfill \end{array}$$

(4)

with Q denoting the system configuration at time t. A similar expansion in powers of $\Delta t$ is not possible for $q_t$ (or $p_t$) due to its rapid fluctuations. Likewise, $P_t$ cannot be directly expanded in $\Delta t$ because it is governed by interactions with the fast bath degrees of freedom (see Equation (3c)).

2.2. Statistical Mechanics

The total energy of the bath and system together,

$$E_{\mathrm{tot}}\left(\lambda \right)=H_{\mathrm{bulk}}\left(\varphi \right)+H_{\mathrm{int}}(q,Q)+H_{\mathrm{sys}}(\mathrm{\Phi },\lambda ),$$

(5)

is conserved by the dynamics (2) for fixed $\lambda$. In the general case of external time-dependent protocols $\lambda =\lambda \left(t\right)$, the energy is not conserved. However, only slow changes in $\lambda \left(t\right)$ over mesoscopic time intervals $\Delta t\gg \tau$ are permitted (and relevant experimentally). From the perspective of the bath, variations in $\lambda$ are thus quasi-statically slow, such that the bath has a well-defined equilibrium energy within each time interval $\Delta t$. This energy is determined by the current values of $\mathrm{\Phi }$,

$$\begin{array}{cc}\hfill E=E\left(\mathrm{\Phi }\right)& =E_{\mathrm{tot}}\left(\lambda \right)-H_{\mathrm{sys}}(\mathrm{\Phi },\lambda )\hfill \\ \hfill & =H_{\mathrm{bulk}}\left(\varphi \right)+H_{\mathrm{int}}(q,Q)\hfill \\ \hfill & =H_{\mathrm{bath}}(\varphi ,Q).\hfill \end{array}$$

(6)

Note that if all parameters $(\varphi ,\mathrm{\Phi })$ remain fixed and only $\lambda$ is varied, the total energy is changed via the contribution in the system potential, while the bath energy remains unchanged (and therefore does not directly depend on $\lambda$).

According to our equilibrium assumption, the bath can be described by the microcanonical ensemble, which is consistent with the current values of $\mathrm{\Phi }$ and $\lambda$. The corresponding microcanonical partition function $\mathrm{\Omega }\left(E\right)$ is given by [8]

$$\mathrm{\Omega }\left(E\right)=\mathrm{\Omega }\left(E\left(\mathrm{\Phi }\right)\right)=\int \mathrm{d}\varphi \delta \left(E\left(\mathrm{\Phi }\right)-H_{\mathrm{bath}}(\varphi ,Q)\right).$$

(7)

According to standard statistical mechanics [8], the microcanonical density is

$${\rho }_{\mathrm{mc}}\left(\varphi \right)=\frac{\delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)}{\mathrm{\Omega }\left(E\right)},$$

(8)

and the temperature T of the bath is defined as ($k_{\mathrm{B}}$ is Boltzmann’s constant)

$$\frac{1}{k_{\mathrm{B}}T}=\frac{\partial }{\partial E}ln\mathrm{\Omega }\left(E\right)=\frac{{\mathrm{\Omega }}^{\prime }\left(E\right)}{\mathrm{\Omega }\left(E\right)}.$$

(9)

3. Effective System Dynamics

Starting from an initial state of the bath and system together, the specific evolution of the system state $\mathrm{\Phi }$ according to (2c), (2d) is affected by the rapidly fluctuating bath degrees of freedom, uniquely emerging from their initial state (and the one of the system). Different initial states of the bath thus lead to different “realizations” of the system dynamics, even if they start from the same initial configuration. Since we do not know the initial state of the bath exactly, but only that it is consistent with the microcanonical density (8), we can write the transition probability density for the system to evolve from $\mathrm{\Phi }=(P,Q)$ to ${\mathrm{\Phi }}^{\prime }=(P^{\prime },Q^{\prime })$ during a mesoscopic time interval $\Delta t$ as

$$p_E\left({\mathrm{\Phi }}^{\prime }\right|\mathrm{\Phi })=\frac{\int \mathrm{d}\varphi \delta (E-H_{\mathrm{bath}}(\varphi ,Q))\delta ({\mathrm{\Phi }}^{\prime }-{\mathrm{\Phi }}_{\Delta t}(\varphi ,\mathrm{\Phi }))}{\mathrm{\Omega }\left(E\right)},$$

(10)

where we denote the initial state of the bath at the beginning of the time interval by the unprimed variable $\varphi$. This probability density gives rise to the statistical properties of “small displacements” $\Delta \mathrm{\Phi }={\mathrm{\Phi }}^{\prime }-\mathrm{\Phi }$ of the system degrees of freedom. Their analysis is the central aim of this paper. Before evaluating averages and (co-)variances in Section 3.2, we start in Section 3.1 with demonstrating that the system displacements within successive time steps $\Delta t$ are independent.

3.1. Markov Property

We consider a configuration $(\varphi ,\mathrm{\Phi })$ of the bath and system together, which, from time t, evolves for a duration $2\Delta t$ according to the full microscopic dynamics (2). Generalizing (10), the probability density for finding the system in configuration ${\mathrm{\Phi }}^{\prime }$ at time $t+\Delta t$ and in configuration ${\mathrm{\Phi }}^{″}$ at time $t+2\Delta t$ when it started at $\mathrm{\Phi }$ at time t is

$$\begin{array}{cc}\hfill & p({\mathrm{\Phi }}^{″},t+2\Delta t;{\mathrm{\Phi }}^{\prime },t+\Delta t|\mathrm{\Phi },t)\hfill \\ \hfill & =\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\delta \left({\mathrm{\Phi }}^{\prime }-{\mathrm{\Phi }}_{\Delta t}(\varphi ,\mathrm{\Phi })\right)\delta \left({\mathrm{\Phi }}^{″}-{\mathrm{\Phi }}_{2\Delta t}(\varphi ,\mathrm{\Phi })\right).\hfill \end{array}$$

(11)

We will show in the following that $p({\mathrm{\Phi }}^{″},t+2\Delta t;{\mathrm{\Phi }}^{\prime },t+\Delta t|\mathrm{\Phi },t)$ fulfills the Markov property

$$p({\mathrm{\Phi }}^{″},t+2\Delta t;{\mathrm{\Phi }}^{\prime },t+\Delta t|\mathrm{\Phi },t)=p({\mathrm{\Phi }}^{″},t+2\Delta t|{\mathrm{\Phi }}^{\prime },t+\Delta t)p({\mathrm{\Phi }}^{\prime },t+\Delta t|\mathrm{\Phi },t),$$

(12)

where

$$\begin{array}{cc}\hfill p({\mathrm{\Phi }}^{\prime },t+\Delta t|\mathrm{\Phi },t)& =\frac{\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\delta \left({\mathrm{\Phi }}^{\prime }-{\mathrm{\Phi }}_{\Delta t}(\varphi ,\mathrm{\Phi })\right)}{\mathrm{\Omega }\left(E\right)},\hfill \end{array}$$

(13a)

$$\begin{array}{cc}\hfill p({\mathrm{\Phi }}^{″},t+2\Delta t|{\mathrm{\Phi }}^{\prime },t+\Delta t)& =\frac{\int \mathrm{d}{\varphi }^{\prime }\delta \left(E^{\prime }-H_{\mathrm{bath}}({\varphi }^{\prime },Q^{\prime })\right)\delta \left({\mathrm{\Phi }}^{″}-{\mathrm{\Phi }}_{\Delta t}({\varphi }^{\prime },{\mathrm{\Phi }}^{\prime })\right)}{\mathrm{\Omega }\left(E^{\prime }\right)},\hfill \end{array}$$

(13b)

in analogy to (10) (we skip the energy subscript at p for notational convenience). The energy $E^{\prime }$ in (13b) is given by the energy balance (6) for the primed variables, and with $\lambda =\lambda (t+\Delta t)$.

Starting from the expression (11), we first use the composition property of solutions to the Hamiltonian equations of motion to write $\delta \left({\mathrm{\Phi }}^{″}-{\mathrm{\Phi }}_{2\Delta t}(\varphi ,\mathrm{\Phi })\right)=\delta \left({\mathrm{\Phi }}^{″}-{\mathrm{\Phi }}_{\Delta t}({\varphi }_{\Delta t}(\varphi ,\mathrm{\Phi }),{\mathrm{\Phi }}_{\Delta t}(\varphi ,\mathrm{\Phi }))\right)$. Then, we insert unity into the integral, expressed in the form $\int \mathrm{d}{\varphi }^{\prime }\delta \left({\varphi }^{\prime }-{\varphi }_{\Delta t}(\varphi ,\mathrm{\Phi })\right)$. With the presence of this delta function in the integrand, we can replace ${\varphi }_{\Delta t}(\varphi ,\mathrm{\Phi })$ everywhere by ${\varphi }^{\prime }$, and likewise for ${\mathrm{\Phi }}^{\prime }$ due to the delta function $\delta \left({\mathrm{\Phi }}^{\prime }-{\mathrm{\Phi }}_{\Delta t}(\varphi ,\mathrm{\Phi })\right)$. These steps turn (11) into

$$\begin{array}{cc}\hfill & p({\mathrm{\Phi }}^{″},t+2\Delta t;{\mathrm{\Phi }}^{\prime },t+\Delta t|\mathrm{\Phi },t)\hfill \\ \hfill & =\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\delta \left({\mathrm{\Phi }}^{\prime }-{\mathrm{\Phi }}_{\Delta t}(\varphi ,\mathrm{\Phi })\right)\hfill \\ \hfill & \times \int \mathrm{d}{\varphi }^{\prime }\delta \left({\varphi }^{\prime }-{\varphi }_{\Delta t}(\varphi ,\mathrm{\Phi })\right)\delta \left({\mathrm{\Phi }}^{″}-{\mathrm{\Phi }}_{\Delta t}({\varphi }^{\prime },{\mathrm{\Phi }}^{\prime })\right)\hfill \\ \hfill & =\int \mathrm{d}{\varphi }^{\prime }\delta \left({\mathrm{\Phi }}^{″}-{\mathrm{\Phi }}_{\Delta t}({\varphi }^{\prime },{\mathrm{\Phi }}^{\prime })\right)\hfill \\ \hfill & \times \frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\delta \left({\varphi }^{\prime }-{\varphi }_{\Delta t}(\varphi ,\mathrm{\Phi })\right)\delta \left({\mathrm{\Phi }}^{\prime }-{\mathrm{\Phi }}_{\Delta t}(\varphi ,\mathrm{\Phi })\right),\hfill \end{array}$$

(14)

exchanging the order of integration in the second equality. Here, the second line is the probability density $p({\varphi }^{\prime },{\mathrm{\Phi }}^{\prime },t+\Delta t|\mathrm{\Phi },t)$ for the total system to be in the state $({\varphi }^{\prime },{\mathrm{\Phi }}^{\prime })$ after evolving for time $\Delta t$, given that the system is in state $\mathrm{\Phi }$ at time t. Hence,

$$p({\mathrm{\Phi }}^{″},t+2\Delta t;{\mathrm{\Phi }}^{\prime },t+\Delta t|\mathrm{\Phi },t)=\int \mathrm{d}{\varphi }^{\prime }\delta \left({\mathrm{\Phi }}^{″}-{\mathrm{\Phi }}_{\Delta t}({\varphi }^{\prime },{\mathrm{\Phi }}^{\prime })\right)p({\varphi }^{\prime },{\mathrm{\Phi }}^{\prime },t+\Delta t|\mathrm{\Phi },t).$$

(15)

So far, the calculations are exact. Now, however, we have to make use of our central assumption that the bath is in equilibrium at any time point t, $t+\Delta t$, $t+2\Delta t$, etc., with a bath energy that is consistent with the current state of the system. This implies that the probability density of bath configurations ${\varphi }^{\prime }$ depends only on the system state at the same time point, but not on its state at earlier times (this is used in the second step below), and that it is given by the microcanonical density (used in the third step, cf. Equation (8)). We can thus write

$$\begin{array}{cc}\hfill p({\varphi }^{\prime },{\mathrm{\Phi }}^{\prime },t+\Delta t|\mathrm{\Phi },t)& =p({\varphi }^{\prime },t+\Delta t|{\mathrm{\Phi }}^{\prime },t+\Delta t;\mathrm{\Phi },t)p({\mathrm{\Phi }}^{\prime },t+\Delta t|\mathrm{\Phi },t)\hfill \\ \hfill & =p({\varphi }^{\prime },t+\Delta t|{\mathrm{\Phi }}^{\prime },t+\Delta t)p({\mathrm{\Phi }}^{\prime },t+\Delta t|\mathrm{\Phi },t)\hfill \\ \hfill & =\frac{\delta \left(E^{\prime }-H_{\mathrm{bath}}({\varphi }^{\prime },Q^{\prime })\right)}{\mathrm{\Omega }\left(E^{\prime }\right)}p({\mathrm{\Phi }}^{\prime },t+\Delta t|\mathrm{\Phi },t).\hfill \end{array}$$

(16)

Plugging this relation into (15), we obtain

$$\begin{array}{cc}\hfill & p({\mathrm{\Phi }}^{″},t+2\Delta t;{\mathrm{\Phi }}^{\prime },t+\Delta t|\mathrm{\Phi },t)\hfill \\ \hfill & =p({\mathrm{\Phi }}^{\prime },t+\Delta t|\mathrm{\Phi },t)\frac{1}{\mathrm{\Omega }\left(E^{\prime }\right)}\int \mathrm{d}{\varphi }^{\prime }\delta \left(E^{\prime }-H_{\mathrm{bath}}({\varphi }^{\prime },Q^{\prime })\right)\delta \left({\mathrm{\Phi }}^{″}-{\mathrm{\Phi }}_{\Delta t}({\varphi }^{\prime },{\mathrm{\Phi }}^{\prime })\right)\hfill \\ \hfill & =p({\mathrm{\Phi }}^{\prime },t+\Delta t|\mathrm{\Phi },t)p({\mathrm{\Phi }}^{″},t+2\Delta t|{\mathrm{\Phi }}^{\prime },t+\Delta t).\hfill \end{array}$$

(17)

In the second equality, we used the relation (13b), completing the proof of the Markov property (12) for the effective system dynamics on $\Delta t$ time scales.

3.2. Average System Behavior

The main idea is to calculate from (10) the average displacements $\langle {\mathrm{\Phi }}^{\prime }-\mathrm{\Phi }\rangle =\langle \Delta \mathrm{\Phi }\rangle$ and various (co-)variances $\langle \Delta \mathrm{\Phi }\Delta \mathrm{\Phi }\rangle$, in linear order in $\Delta t$, and to assess which kind of stochastic evolution equation for the system dynamics alone would produce the same moments in leading order $\Delta t$. According to our result from the previous section, it is sufficient to consider any such time interval $\Delta t$ because the system dynamics are Markovian.

For clarity, we consider just one system particle such that $\mathrm{\Phi }=(P,Q)=(\mathit{P},\mathit{Q})$ with $P=\mathit{P}=(P_1,P_2,P_3)$ and $Q=\mathit{Q}=(Q_1,Q_2,Q_3)$. We then have to evaluate ($i,j=1,2,3$)

$$\begin{array}{cc}\hfill \langle \Delta Q_i\rangle & =\langle Q_i^{\prime }-Q_i\rangle \hfill \\ \hfill & =\int \mathrm{d}{\mathrm{\Phi }}^{\prime }{p}_E\left({\mathrm{\Phi }}^{\prime }\right|\mathrm{\Phi })(Q_i^{\prime }-Q_i),\hfill \\ \hfill \langle \Delta P_i\rangle & =\langle P_i^{\prime }-P_i\rangle \hfill \end{array}$$

(18a)

$$\begin{array}{cc}\hfill & =\int \mathrm{d}{\mathrm{\Phi }}^{\prime }{p}_E\left({\mathrm{\Phi }}^{\prime }\right|\mathrm{\Phi })(P_i^{\prime }-P_i),\hfill \end{array}$$

(18b)

$$\begin{array}{cc}\hfill \langle \Delta Q_i\Delta Q_j\rangle & =\int \mathrm{d}{\mathrm{\Phi }}^{\prime }{p}_E\left({\mathrm{\Phi }}^{\prime }\right|\mathrm{\Phi })(Q_i^{\prime }-Q_i)(Q_j^{\prime }-Q_j),\hfill \end{array}$$

(18c)

$$\begin{array}{cc}\hfill \langle \Delta P_i\Delta P_j\rangle & =\int \mathrm{d}{\mathrm{\Phi }}^{\prime }{p}_E\left({\mathrm{\Phi }}^{\prime }\right|\mathrm{\Phi })(P_i^{\prime }-P_i)(P_j^{\prime }-P_j),\hfill \end{array}$$

(18d)

$$\begin{array}{cc}\hfill \langle \Delta Q_i\Delta P_j\rangle & =\int \mathrm{d}{\mathrm{\Phi }}^{\prime }{p}_E\left({\mathrm{\Phi }}^{\prime }\right|\mathrm{\Phi })(Q_i^{\prime }-Q_i)(P_j^{\prime }-P_j),\hfill \end{array}$$

(18e)

to lowest order in $\Delta t$. Note that the averages in (18) are over the final configuration ${\mathrm{\Phi }}^{\prime }$ only, conditioned on a fixed (but arbitrary) initial value $\mathrm{\Phi }$.

3.2.1. Evaluation of $\langle \Delta Q_i\rangle$

We use the expression (10) for $p_E\left({\mathrm{\Phi }}^{\prime }\right|\mathrm{\Phi })$ in (18a):

$$\begin{array}{cc}\hfill \langle \Delta Q_i\rangle & =\int \mathrm{d}{\mathrm{\Phi }}^{\prime }(Q_i^{\prime }-Q_i)\frac{\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\delta \left({\mathrm{\Phi }}^{\prime }-{\mathrm{\Phi }}_{\Delta t}(\varphi ,\mathrm{\Phi })\right)}{\mathrm{\Omega }\left(E\right)}\hfill \\ \hfill & =\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \left[{\left(Q_i\right)}_{\Delta t}(\varphi ,\mathrm{\Phi })-Q_i\right]\delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right).\hfill \end{array}$$

(19)

Since ${\left(Q_i\right)}_{\Delta t}(\varphi ,\mathrm{\Phi })$ is a slowly varying function of t on $\Delta t$ time intervals, we can use the expansion from (4) and obtain

$$\begin{array}{cc}\hfill \langle \Delta Q_i\rangle & =\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \left[Q_i+\frac{P_i}{M}\Delta t-Q_i\right]\delta (E-H_{\mathrm{bath}}(\varphi ,Q))+\mathcal{O}(\Delta t^2)\hfill \\ \hfill & =\frac{P_i}{M}\Delta t+\mathcal{O}(\Delta t^2).\hfill \end{array}$$

(20)

The linear order result is therefore, as expected,

$$\langle \Delta Q_i\rangle =\frac{P_i}{M}\Delta t.$$

(21)

3.2.2. Evaluation of $\langle \Delta P_i\rangle$

We start again by using the expression (10) for $p_E\left({\mathrm{\Phi }}^{\prime }\right|\mathrm{\Phi })$,

$$\begin{array}{cc}\hfill \langle \Delta P_i\rangle & =\int \mathrm{d}{\mathrm{\Phi }}^{\prime }(P_i^{\prime }-P_i)\frac{\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\delta \left({\mathrm{\Phi }}^{\prime }-{\mathrm{\Phi }}_{\Delta t}(\varphi ,\mathrm{\Phi })\right)}{\mathrm{\Omega }\left(E\right)}\hfill \\ \hfill & =\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \left[{\left(P_i\right)}_{\Delta t}(\varphi ,\mathrm{\Phi })-P_i\right]\delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\hfill \\ \hfill & =-\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi {\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\left[\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q_i}+\frac{\partial H_{\mathrm{sys}}({\mathrm{\Phi }}_{t^{\prime }},{\lambda }_{t^{\prime }})}{\partial Q_i}\right]\delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right),\hfill \end{array}$$

(22)

where, in the last line, we inserted the formal solution from (3c). The system contribution (second term in the brackets) is a slowly varying function of time on $\Delta t$ scales. Defining the external force on the system,

$$f(Q,\lambda )=-\frac{\partial U(Q,\lambda )}{\partial Q}=-\frac{\partial H_{\mathrm{sys}}(\mathrm{\Phi },\lambda )}{\partial Q},$$

(23)

and expanding $Q_{t^{\prime }}=Q+(P/M)(t^{\prime }-t)+\dots$ as above and ${\lambda }_{t^{\prime }}$ as ${\lambda }_{t^{\prime }}={\lambda }_{t^{\prime }=t}+{\dot{\lambda }}_{t^{\prime }=t}(t^{\prime }-t)+\dots$, we find $-\frac{\partial H_{\mathrm{sys}}({\mathrm{\Phi }}_{t^{\prime }},{\lambda }_{t^{\prime }})}{\partial Q_i}=f_i(Q,\lambda )+\mathcal{O}(t^{\prime }-t)$ (with $\lambda ={\lambda }_{t^{\prime }=t}$). Using this expansion, we can perform the time integral and the microcanonical average for the system contribution. We obtain for (22)

$$\langle \Delta P_i\rangle =f_i(Q,\lambda )\Delta t-\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi {\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q_i}\delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)+\mathcal{O}(\Delta t^2).$$

(24)

A similar expansion procedure is not achievable for the interaction part because $H_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})={\sum }_{\mu }{U}_{\mathrm{int}}\left(|{\left({\mathit{q}}_{\mu }\right)}_{t^{\prime }}-{\mathit{Q}}_{t^{\prime }}|\right)$ is a highly fluctuating function of time. The contribution $Q_{t^{\prime }}\approx Q+(P/M)(t^{\prime }-t)$ just superimposes a slow drift over the rapid fluctuations. Nevertheless, in order to analyze this term, we can start by evaluating the “zeroth-order” effect by approximating $Q_{t^{\prime }}$ with its initial value, $Q_t=Q$. The interaction term is then

$$\begin{array}{c}-\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\int \mathrm{d}\varphi \frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q_i}\delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\hfill \\ \hfill \approx -\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\int \mathrm{d}\varphi \frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right).\end{array}$$

(25)

The $\varphi$ integral

$$\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}={〈\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}〉}_{\mathrm{mc}}$$

(26)

represents the momentary, average net force (component i) that the bath exerts on an immobile system particle. For physical reasons, we expect this force to vanish, as there should be no net force on the particle from the thermal environment. In Appendix A, we prove that this is indeed the case.

The question now arises as to whether there will be a net $\Delta t$ contribution from

$$-\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q_i},$$

(27)

when considering the next order $Q_{t^{\prime }}\approx Q+(P/M)(t^{\prime }-t)$ in the particle displacement. In this case, the particle moves (slowly) relative to the bath and we indeed expect a frictional contribution to show up, which, in lowest order, should be proportional to $(P/M)\Delta t$ (see also the illuminating discussion in Section 15.5 of [8]). Since we cannot expand $\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}$ in $t^{\prime }$, we extract the leading contribution by a partial integration in the time integral. Moreover, from a physical perspective, it will be instructive to transform to coordinates co-moving with the system particle. We will perform the calculation with and without the transformation to the co-moving frame, which will result in two alternative (but equivalent) expressions for the prefactor of $(P/M)\Delta t$.

Co-Moving Frame

In order to switch to the co-moving frame, we introduce for all $\mu =1,2,\dots ,n$ the time-dependent coordinate transformation (for fixed initial time t at which the bath and system are in the state $({\varphi }_t,{\mathrm{\Phi }}_t)=(\varphi ,\mathrm{\Phi })$)

$${\tilde{\mathit{q}}}_{\mu }\left(t^{\prime }\right)={\mathit{q}}_{\mu }-({\mathit{Q}}_{t^{\prime }}-\mathit{Q})\approx {\mathit{q}}_{\mu }-\frac{\mathit{P}}{M}(t^{\prime }-t)\iff {\mathit{q}}_{\mu }={\tilde{\mathit{q}}}_{\mu }\left(t^{\prime }\right)+\frac{\mathit{P}}{M}(t^{\prime }-t).$$

(28)

Differences between bath particle coordinates are invariant under this transformation, implying that

$$H_{\mathrm{bulk}}(p,q)=H_{\mathrm{bulk}}(p,\tilde{q}\left(t\right)).$$

(29a)

Differences between bath particle coordinates and the system coordinate transform according to

$$\begin{array}{cc}\hfill {\mathit{q}}_{\mu }-\mathit{Q}& ={\tilde{\mathit{q}}}_{\mu }\left(t^{\prime }\right)-\left(\mathit{Q}-\frac{\mathit{P}}{M}(t^{\prime }-t)\right),\hfill \\ \hfill {\left({\mathit{q}}_{\mu }\right)}_{t^{\prime }}-{\mathit{Q}}_{t^{\prime }}& ={\mathit{q}}_{\mu }+{\int }_t^{t^{\prime }}\mathrm{d}{t}^{″}\frac{{\left({\mathit{p}}_{\mu }\right)}_{t^{″}}}{m}-\left(\mathit{Q}+\frac{\mathit{P}}{M}(t^{\prime }-t)\right)\hfill \end{array}$$

(29b)

$$\begin{array}{cc}\hfill & ={\tilde{\mathit{q}}}_{\mu }\left(t^{\prime }\right)+{\int }_t^{t^{\prime }}\mathrm{d}{t}^{″}\frac{{\left({\mathit{p}}_{\mu }\right)}_{t^{″}}}{m}-\mathit{Q}.\hfill \end{array}$$

(29c)

Here, we use the formal solutions (3b) and (3d) (see also (4)). We moreover note that the solution (3a) for ${\mathit{p}}_{\mu }$, i.e., ${\left({\mathit{p}}_{\mu }\right)}_{t^{\prime }}={\mathit{p}}_{\mu }-{\int }_t^{t^{\prime }}\mathrm{d}{t}^{″}\frac{\partial H_{\mathrm{bath}}({\varphi }_{t^{″}},Q_{t^{″}})}{\partial {\mathit{q}}_{\mu }}$, contains the transformed ${\tilde{\mathit{q}}}_{\mu }\left(t^{\prime }\right)={\mathit{q}}_{\mu }-\frac{\mathit{P}}{M}(t^{\prime }-t)$ in $\frac{\partial H_{\mathrm{int}}(q_{t^{″}},Q_{t^{″}})}{\partial {\mathit{q}}_{\mu }}$ via its dependence on the differences ${\mathit{q}}_{\mu }-\mathit{Q}$, and analogously for similar expressions. The transformation to the co-moving frame can therefore be seen as a time-dependent shift $q\to \tilde{q}\left(t^{\prime }\right)$ in the initial q value. Since the integral $\int \mathrm{d}q$ in (27) is over all possible q values, and since $\mathrm{d}q=\mathrm{d}\tilde{q}\left(t^{\prime }\right)$, this is the same for all transformations $\tilde{q}\left(t^{\prime }\right)$, independent of the time point $t^{\prime }$. Within the integral, $\tilde{q}\left(t^{\prime }\right)$ therefore plays the role of a new initial value, i.e., under the integral, we can drop the time argument in $\tilde{q}\left(t^{\prime }\right)$. In the following calculation (second equal sign below), we make this explicit by replacing $\tilde{q}\left(t^{\prime }\right)$ with the symbol q, which we used to indicate the initial values at starting time t all along. Using the transformation (28), we apply (29) to rewrite the interaction part (27) as

$$\begin{array}{cc}\hfill & -\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q_i}\hfill \\ \hfill & = -\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\int \mathrm{d}p\mathrm{d}\tilde{q}\left(t^{\prime }\right)\hfill \\ \hfill & \times \delta \left(E-H_{\mathrm{bulk}}(p,\tilde{q}\left(t^{\prime }\right))-\sum _{\mu }{U}_{\mathrm{int}}\left(|{\tilde{\mathit{q}}}_{\mu }\left(t^{\prime }\right)-[\mathit{Q}-\frac{\mathit{P}}{M}(t^{\prime }-t)]|\right)\right)\hfill \\ \hfill & \times \sum _{\mu }\frac{\partial U_{\mathrm{int}}\left(\right|{\tilde{\mathit{q}}}_{\mu }\left(t^{\prime }\right)+{\int }_t^{t^{\prime }}\mathrm{d}{t}^{″}\frac{{\left({\mathit{p}}_{\mu }\right)}_{t^{\prime \prime }}}{m}-\mathit{Q}\left|\right)}{\partial Q_i}\hfill \\ \hfill & = -\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\int \mathrm{d}p\mathrm{d}q\delta \left(E-H_{\mathrm{bulk}}(p,q)-\sum _{\mu }{U}_{\mathrm{int}}\left(|{\mathit{q}}_{\mu }-[\mathit{Q}-\frac{\mathit{P}}{M}(t^{\prime }-t)]|\right)\right)\hfill \\ \hfill & \times \sum _{\mu }\frac{\partial U_{\mathrm{int}}\left(\right|{\mathit{q}}_{\mu }+{\int }_t^{t^{\prime }}\mathrm{d}{t}^{″}\frac{{\left({\mathit{p}}_{\mu }\right)}_{t^{\prime \prime }}}{m}-\mathit{Q}\left|\right)}{\partial Q_i}\hfill \\ \hfill & = -\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bulk}}\left(\varphi \right)-H_{\mathrm{int}}(q,Q-\frac{P}{M}[t^{\prime }-t])\right)\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}.\hfill \end{array}$$

(30)

The main effect of the transformation to the co-moving frame is thus to move the time dependence in $Q_{t^{\prime }}$ from the interaction force outside the $\delta$ function to the interaction Hamiltonian inside the $\delta$ function.

In order to extract the leading order in $\Delta t$, we introduce $1=-\frac{\partial }{\partial t^{\prime }}[\Delta t-(t^{\prime }-t)]$ in the time integral and perform a partial integration (using the “more natural” insertion $1=\frac{\partial }{\partial t^{\prime }}(t^{\prime }-t)$ rather than $1=-\frac{\partial }{\partial t^{\prime }}[\Delta t-(t^{\prime }-t)]$ would lead to non-vanishing boundary terms after partial integration):

$$\begin{array}{cc}\hfill & -\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bulk}}\left(\varphi \right)-H_{\mathrm{int}}(q,Q-\frac{P}{M}(t^{\prime }-t))\right)\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\hfill \\ \hfill & = -\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\left[-\frac{\partial }{\partial t^{\prime }}[\Delta t-(t^{\prime }-t)]\right]\hfill \\ \hfill & \times \int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bulk}}\left(\varphi \right)-H_{\mathrm{int}}(q,Q-\frac{P}{M}(t^{\prime }-t))\right)\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\hfill \\ \hfill & = \frac{1}{\mathrm{\Omega }\left(E\right)}{\left[[\Delta t-(t^{\prime }-t)]\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bulk}}\left(\varphi \right)-H_{\mathrm{int}}(q,Q-\frac{P}{M}(t^{\prime }-t))\right)\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\right]}_{t^{\prime }=t}^{t+\Delta t}\hfill \\ \hfill & -\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }[\Delta t-(t^{\prime }-t)]\int \mathrm{d}\varphi {\delta }^{\prime }\left(E-H_{\mathrm{bulk}}\left(\varphi \right)-H_{\mathrm{int}}(q,Q-\frac{P}{M}(t^{\prime }-t))\right)\hfill \\ \hfill & \times \sum _j\left[-\frac{\partial H_{\mathrm{int}}(q,Q-\frac{P}{M}(t^{\prime }-t)))}{\partial Q_j}\left(-\frac{P_j}{M}\right)\right]\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\hfill \\ \hfill & -\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }[\Delta t-(t^{\prime }-t)]\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bulk}}\left(\varphi \right)-H_{\mathrm{int}}(q,Q-\frac{P}{M}(t^{\prime }-t))\right)\hfill \\ \hfill & \times \sum _{\mu ,j}\frac{{\partial }^2{H}_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial q_{\mu ,j}\partial Q_i}{\left({\dot{q}}_{\mu ,j}\right)}_{t^{\prime }}.\hfill \end{array}$$

(31)

The first line in the last equality is the boundary term from the partial integration. In addition, the partial integration produces two new terms (second and third summand) due to the (slow) time dependence in the interaction Hamiltonian within the $\delta$ function and the (fast) one in the interaction force outside the $\delta$ function.

Next, we transform back to the original coordinates in the laboratory-fixed frame (and use $q_t=q$, $Q_t=Q$ again):

$$\begin{array}{cc}\hfill & -\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bulk}}\left(\varphi \right)-H_{\mathrm{int}}(q,Q-\frac{P}{M}(t^{\prime }-t))\right)\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\hfill \\ \hfill & = -\frac{\Delta t}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bulk}}\left(\varphi \right)-H_{\mathrm{int}}(q,Q)\right)\frac{\partial H_{\mathrm{int}}(q,Q)}{\partial Q_i}\hfill \\ \hfill & -\sum _j\frac{P_j}{M}\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }[\Delta t-(t^{\prime }-t)]\hfill \\ \hfill & \times \int \mathrm{d}\varphi {\delta }^{\prime }\left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\frac{\partial H_{\mathrm{int}}(q,Q)}{\partial Q_j}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q_i}\hfill \\ \hfill & -\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }[\Delta t-(t^{\prime }-t)]\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\sum _{\mu ,j}\frac{{\partial }^2{H}_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial q_{\mu ,j}\partial Q_i}\frac{{\left(p_{\mu ,j}\right)}_{t^{\prime }}}{m}.\hfill \end{array}$$

(32)

The first line corresponds to the zeroth-order contribution discussed above (see Equations (25) and (26)) and is proven in Appendix A to vanish for homogeneous, translational-invariant environments (in other cases, this term would result in a net force from a so-called ”potential of mean force” [9]). Likewise, we show in Appendix A that phase-space averages of the type like the one in the last line vanish in zeroth order $\Delta t$, for which $Q_{t^{\prime }}\approx Q_t=Q$ (see Equation (A7)), implying that they contribute only in $\mathcal{O}(\Delta t^2)$. Hence, the only contribution in linear order $\Delta t$ comes from the term in the second and third line when approximating $Q_{t^{\prime }}\approx Q$. The prime at the $\delta$ function denotes the derivative with respect to its argument, or, equivalently, with respect to E. Using the definition of temperature (9), we prove in Appendix B that, for “non-extensive” averages (like averages over short-ranged interactions between the system particle and bath), we are allowed to replace ${\delta }^{\prime }(\dots )$ with $\frac{1}{k_{\mathrm{B}}T}\delta (\dots )$. We finally find that

$$\begin{array}{cc}\hfill & -\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q_i}\hfill \\ \hfill & = -\sum _j\frac{P_j}{M}\frac{1}{k_{\mathrm{B}}T}\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}t[\Delta t-(t^{\prime }-t)]\hfill \\ \hfill & \times \int \mathrm{d}\varphi \delta (E-H_{\mathrm{bath}}(\varphi ,Q))\frac{\partial H_{\mathrm{int}}(q,Q)}{\partial Q_j}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}+\mathcal{O}(\Delta t^2)\hfill \\ \hfill & = -\sum _j\frac{P_j}{M}\Delta t\frac{1}{k_{\mathrm{B}}T}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\left(1-\frac{t^{\prime }-t}{\Delta t}\right){〈\frac{\partial H_{\mathrm{int}}(q,Q)}{\partial Q_j}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}〉}_{\mathrm{mc}}+\mathcal{O}(\Delta t^2).\hfill \end{array}$$

(33)

Laboratory Frame

We can, of course, perform exactly the same partial time integration as in the calculation above without transforming to the co-moving frame. Using again the results (A7) from Appendix A, which imply that certain terms do not contribute in linear order $\Delta t$ (in the third and fourth step below), the main calculation steps are

$$\begin{array}{cc}\hfill & -\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q_i}\hfill \\ \hfill & = -\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\left[-\frac{\partial }{\partial t^{\prime }}[\Delta t-(t^{\prime }-t)]\right]\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q_i}\hfill \\ \hfill & = \frac{1}{\mathrm{\Omega }\left(E\right)}{\left[[\Delta t-(t^{\prime }-t)]\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q_i}\right]}_{t^{\prime }=t}^{t+\Delta t}\hfill \\ \hfill & -\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }[\Delta t-(t^{\prime }-t)]\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\frac{\partial }{\partial t^{\prime }}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q_{{ }^{\prime }t})}{\partial Q_i}\hfill \\ \hfill & = \mathcal{O}(\Delta t^2)-\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }[\Delta t-(t^{\prime }-t)]\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\hfill \\ \hfill & \times \left[\sum _j\frac{{\partial }^2{H}_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q_j\partial Q_i}\frac{P_j}{M}+\sum _{\mu ,j}\frac{{\partial }^2{H}_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial q_{\mu ,j}\partial Q_i}\frac{{\left(p_{\mu ,j}\right)}_{t^{\prime }}}{m}\right]\hfill \\ \hfill & = -\sum _j\frac{P_j}{M}\Delta t{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\left(1-\frac{t^{\prime }-t}{\Delta t}\right)\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\frac{{\partial }^2{H}_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q_j\partial Q_i}+\mathcal{O}(\Delta t^2).\hfill \end{array}$$

(34)

Finally,

$$\begin{array}{cc}\hfill & -\frac{1}{\mathrm{\Omega }\left(E\right)}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q_i}\hfill \\ \hfill & = -\sum _j\frac{P_j}{M}\Delta t{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\left(1-\frac{t^{\prime }-t}{\Delta t}\right){〈\frac{{\partial }^2{H}_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_j\partial Q_i}〉}_{\mathrm{mc}}+\mathcal{O}(\Delta t^2).\hfill \end{array}$$

(35)

The $\mathcal{O}(\Delta t)$ Contribution in $\langle \Delta P_i\rangle$

Summarizing what we found for $\langle \Delta P_i\rangle$, the contribution to linear order in $\Delta t$ is

$$\langle \Delta P_i\rangle =f_i(Q,\lambda )\Delta t-\sum _j{\gamma }_{ij}\frac{P_j}{M}\Delta t.$$

(36)

Here, we introduce the friction coefficient ${\gamma }_{ij}$, given by the two alternative expressions from (33) and (35):

$$\begin{array}{cc}\hfill {\gamma }_{ij}& =\frac{1}{k_{\mathrm{B}}T}{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\left(1-\frac{t^{\prime }-t}{\Delta t}\right){〈\frac{\partial H_{\mathrm{int}}(q,Q)}{\partial Q_j}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}〉}_{\mathrm{mc}}\hfill \\ \hfill & ={\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\left(1-\frac{t^{\prime }-t}{\Delta t}\right){〈\frac{{\partial }^2{H}_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_j\partial Q_i}〉}_{\mathrm{mc}},\hfill \end{array}$$

(37)

We remark that the equivalence of $\frac{1}{k_{\mathrm{B}}T}{〈\frac{\partial H_{\mathrm{int}}(q,Q)}{\partial Q_j}\frac{\partial H_{\mathrm{int}}(q_t,Q)}{\partial Q_i}〉}_{\mathrm{mc}}$ and ${〈\frac{{\partial }^2{H}_{\mathrm{int}}(q_t,Q)}{\partial Q_j\partial Q_i}〉}_{\mathrm{mc}}$ can be verified directly; see Appendix C. It is obvious from the second line that ${\gamma }_{ij}$ is symmetric under exchange of the indices i,j, a property that is then also valid for the auto-correlation in the first line.

The first expression represents a time integral over the correlation of the interaction forces, averaged over the various dynamical evolutions of the bath particles in the microcanonical ensemble. By shifting the integration variable $t^{\prime }$ to the new variable $\tilde{t}=t^{\prime }-t$ and by defining ${\tilde{q}}_{\tilde{t}}=q_{\tilde{t}+t}$ (such that ${\tilde{q}}_0=q_t=q$), the time integration runs over the interval $[0,\Delta t]$,

$$\begin{array}{cc}\hfill {\gamma }_{ij}& =\frac{1}{k_{\mathrm{B}}T}{\int }_0^{\Delta t}\mathrm{d}t\left(1-\frac{t}{\Delta t}\right){〈\frac{\partial H_{\mathrm{int}}(q,Q)}{\partial Q_j}\frac{\partial H_{\mathrm{int}}(q_t,Q)}{\partial Q_i}〉}_{\mathrm{mc}},\hfill \end{array}$$

(38)

where we drop the tilde symbol to simplify notation. Since we assume that perturbations in the bath decay on the characteristic time scale $\tau$, we expect the correlations to also decay on this time scale, such that ${〈\frac{\partial H_{\mathrm{int}}(q,Q)}{\partial Q_j}\frac{\partial H_{\mathrm{int}}(q_t,Q)}{\partial Q_i}〉}_{\mathrm{mc}}$ vanishes for times $t\gg \tau$. As a consequence, we can extend the upper integration limit $\Delta t\gg \tau$ to infinity. Moreover, the term $t/\Delta t$ in the integral is effectively $\mathcal{O}(\tau /\Delta t)$ and thus negligibly small, as can be verified by rescaling time according to $\tilde{t}=t/\tau$ (to make $\tilde{t}\sim \mathcal{O}\left(1\right)$ when $t\sim \mathcal{O}\left(\tau \right)$). Hence, our expression for the friction tensor becomes

$${\gamma }_{ij}=\frac{1}{k_{\mathrm{B}}T}{\int }_0^{\infty }\mathrm{d}t{〈\frac{\partial H_{\mathrm{int}}(q,Q)}{\partial Q_j}\frac{\partial H_{\mathrm{int}}(q_t,Q)}{\partial Q_i}〉}_{\mathrm{mc}}.$$

(39)

This is exactly the same result as obtained from projection operator techniques; see, e.g., Chapter 11 in [2].

3.2.3. Evaluation of $\langle \Delta Q_i\Delta Q_j\rangle$

The evaluation of $\langle \Delta Q_i\Delta Q_j\rangle$ proceeds along similar lines as the one of $\langle \Delta Q_i\rangle$. We use (10) in (18c) to write

$$\begin{array}{cc}\hfill \langle \Delta Q_i\Delta Q_j\rangle & =\int \mathrm{d}{\mathrm{\Phi }}^{\prime }(Q_i^{\prime }-Q_i)(Q_j^{\prime }-Q_j)\frac{\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\delta \left({\mathrm{\Phi }}^{\prime }-{\mathrm{\Phi }}_{\Delta t}(\varphi ,\mathrm{\Phi })\right)}{\mathrm{\Omega }\left(E\right)}\hfill \\ \hfill & =\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \left[{\left(Q_i\right)}_{\Delta t}(\varphi ,\mathrm{\Phi })-Q_i\right]\left[{\left(Q_j\right)}_{\Delta t}(\varphi ,\mathrm{\Phi })-Q_j\right]\delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right).\hfill \end{array}$$

(40)

For ${\left(Q_i\right)}_{\Delta t}(\varphi ,\mathrm{\Phi })$, we can again use the expansion ${\left(Q_i\right)}_{\Delta t}=Q_i+\frac{P_i}{M}\Delta t+\mathcal{O}(\Delta t^2)$ (see (4)) so that we obtain

$$\begin{array}{cc}\hfill \langle \Delta Q_i\Delta Q_j\rangle & =\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \left(\frac{P_i}{M}\Delta t\right)\left(\frac{P_j}{M}\Delta t\right)\delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)+\mathcal{O}(\Delta t^2)\hfill \\ \hfill & =\mathcal{O}(\Delta t^2).\hfill \end{array}$$

(41)

As expected, $\langle \Delta Q_i\Delta Q_j\rangle$ does not contribute in linear order in $\Delta t$.

3.2.4. Evaluation of $\langle \Delta P_i\Delta P_j\rangle$

In order to evaluate $\langle \Delta P_i\Delta P_j\rangle$, we proceed analogously. We start from (18c) and plug in the transition probability (10) and then the formal solution (3c):

$$\begin{array}{cc}\hfill \langle \Delta P_i\Delta P_j\rangle & =\int \mathrm{d}{\mathrm{\Phi }}^{\prime }(P_i^{\prime }-P_i)(P_j^{\prime }-P_j)\frac{\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\delta \left({\mathrm{\Phi }}^{\prime }-{\mathrm{\Phi }}_{\Delta t}(\varphi ,\mathrm{\Phi })\right)}{\mathrm{\Omega }\left(E\right)}\hfill \\ \hfill & =\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \left[{\left(P_i\right)}_{\Delta t}(\varphi ,\mathrm{\Phi })-P_i\right]\left[{\left(P_j\right)}_{\Delta t}(\varphi ,\mathrm{\Phi })-P_j\right]\delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\hfill \\ \hfill & =\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\left[-{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\left(\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q_i}+\frac{\partial H_{\mathrm{sys}}({\mathrm{\Phi }}_{t^{\prime }},{\lambda }_{t^{\prime }})}{\partial Q_i}\right)\right]\hfill \\ \hfill & \times \left[-{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\left(\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q_j}+\frac{\partial H_{\mathrm{sys}}({\mathrm{\Phi }}_{t^{\prime }},{\lambda }_{t^{\prime }})}{\partial Q_j}\right)\right]\hfill \end{array}$$

(42)

The terms involving $H_{\mathrm{sys}}({\mathrm{\Phi }}_{t^{\prime }},{\lambda }_{t^{\prime }})$ represent the external forces (23) exerted on the system. Recalling that these vary slowly in time on $\Delta t$ intervals, we obtain

$$\begin{array}{cc}\hfill \langle \Delta P_i\Delta P_j\rangle & =-\frac{1}{\mathrm{\Omega }\left(E\right)}{f}_i(Q,\lambda )\Delta t{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_j}\hfill \\ \hfill & -\frac{1}{\mathrm{\Omega }\left(E\right)}{f}_j(Q,\lambda )\Delta t{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\hfill \\ \hfill & +\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\hfill \\ \hfill & \times \left[{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\right]\left[{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_j}\right]\hfill \\ \hfill & +\mathcal{O}(\Delta t^2)\hfill \end{array}$$

(43)

The microcanonical averages appearing in the first two lines are the average net forces of the bath on an immobile system particle, as in (26), and are proven to vanish in Appendix A, cf. Equation (A7a). We thus obtain

$$\langle \Delta P_i\Delta P_j\rangle ={〈{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }{\int }_t^{t+\Delta t}\mathrm{d}{t}^{″}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q_{t^{″}},Q)}{\partial Q_j}〉}_{\mathrm{mc}}+\mathcal{O}(\Delta t^2),$$

(44)

In order to find out if and how this double time integral is related to the friction tensor (37), our aim is to rewrite it as a single time integral. We first split up the square integration domain into two triangular domains and then change the order of integration in one of the domains:

$$\begin{array}{cc}\hfill & {〈{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }{\int }_t^{t+\Delta t}\mathrm{d}{t}^{″}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime \prime }},Q)}{\partial Q_j}〉}_{\mathrm{mc}}\hfill \\ \hfill & ={\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }{\int }_t^{t^{\prime }}\mathrm{d}{t}^{″}{〈\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime \prime }},Q)}{\partial Q_j}〉}_{\mathrm{mc}}\hfill \\ \hfill & +{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }{\int }_{t^{\prime }}^{t+\Delta t}\mathrm{d}{t}^{″}{〈\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime \prime }},Q)}{\partial Q_j}〉}_{\mathrm{mc}}\hfill \\ \hfill & ={\int }_t^{t+\Delta t}\mathrm{d}{t}^{″}{\int }_{t^{″}}^{t+\Delta t}\mathrm{d}{t}^{\prime }{〈\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime \prime }},Q)}{\partial Q_j}〉}_{\mathrm{mc}}\hfill \\ \hfill & +{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }{\int }_{t^{\prime }}^{t+\Delta t}\mathrm{d}{t}^{″}{〈\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime \prime }},Q)}{\partial Q_j}〉}_{\mathrm{mc}}\hfill \\ \hfill & ={\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }{\int }_{t^{\prime }}^{t+\Delta t}\mathrm{d}{t}^{″}[{〈\frac{\partial H_{\mathrm{int}}(q_{t^{″}},Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_j}〉}_{\mathrm{mc}}\hfill \\ \hfill & +{〈\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime \prime }},Q)}{\partial Q_j}〉}_{\mathrm{mc}}].\hfill \end{array}$$

(45)

The last equality is obtained by renaming the integration variables $t^{\prime }\leftrightarrow t^{″}$ in the first term. In the next step, we shift the integration interval in the inner integral by introducing the new variable $\tilde{t}=t^{″}-(t^{\prime }-t)$:

$$\begin{array}{cc}\hfill & {〈{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }{\int }_t^{t+\Delta t}\mathrm{d}{t}^{″}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime \prime }},Q)}{\partial Q_j}〉}_{\mathrm{mc}}\hfill \\ \hfill & ={\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }{\int }_t^{t+\Delta t-(t^{\prime }-t)}\mathrm{d}\tilde{t}[{〈\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }+(\tilde{t}-t)},Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_j}〉}_{\mathrm{mc}}\hfill \\ \hfill & +{〈\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }+(\tilde{t}-t)},Q)}{\partial Q_j}〉}_{\mathrm{mc}}].\hfill \end{array}$$

(46)

The correlations inside the integrals are now expressed in terms of the bath states at time $t^{\prime }$ and at the later time $t^{\prime }+(\tilde{t}-t)$. Due to our central assumption that the bath is in equilibrium, these correlations are functions of the time difference only, i.e., they depend on $\tilde{t}-t$ but not $t^{\prime }$. We are thus free to choose any time point for $t^{\prime }$, and we fix it to the initial time t of the considered time interval $\Delta t$ (remember that the initial bath state is $q_t=q$):

$$\begin{array}{cc}\hfill & {〈{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }{\int }_t^{t+\Delta t}\mathrm{d}{t}^{″}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime \prime }},Q)}{\partial Q_j}〉}_{\mathrm{mc}}\hfill \\ \hfill & ={\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }{\int }_t^{t+\Delta t-(t^{\prime }-t)}\mathrm{d}\tilde{t}[{〈\frac{\partial H_{\mathrm{int}}(q_{\tilde{t}},Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q,Q)}{\partial Q_j}〉}_{\mathrm{mc}}\hfill \\ \hfill & +{〈\frac{\partial H_{\mathrm{int}}(q,Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q_{\tilde{t}},Q)}{\partial Q_j}〉}_{\mathrm{mc}}].\hfill \end{array}$$

(47)

Exploiting the $i\leftrightarrow j$ symmetry of the correlations (cf. Equation (37)), the two terms can now be combined into one. Moreover, since the correlations no longer depend on $t^{\prime }$, we can perform the $t^{\prime }$ integral by exchanging the order of integration:

$$\begin{array}{cc}\hfill & {〈{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }{\int }_t^{t+\Delta t}\mathrm{d}{t}^{″}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime \prime }},Q)}{\partial Q_j}〉}_{\mathrm{mc}}\hfill \\ \hfill & =2{\int }_t^{t+\Delta t}\mathrm{d}\tilde{t}{\int }_t^{t+\Delta t-(\tilde{t}-t)}\mathrm{d}{t}^{\prime }{〈\frac{\partial H_{\mathrm{int}}(q_{\tilde{t}},Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q,Q)}{\partial Q_j}〉}_{\mathrm{mc}}\hfill \\ \hfill & =2{\int }_t^{t+\Delta t}\mathrm{d}\tilde{t}[\Delta t-(\tilde{t}-t)]{〈\frac{\partial H_{\mathrm{int}}(q_{\tilde{t}},Q)}{\partial Q_i}\frac{\partial H_{\mathrm{int}}(q,Q)}{\partial Q_j}〉}_{\mathrm{mc}}.\hfill \end{array}$$

(48)

In linear order in $\Delta t$, we therefore find

$$\begin{array}{cc}\hfill \langle \Delta P_i\Delta P_j\rangle & =2\Delta t{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\left(1-\frac{t^{\prime }-t}{\Delta t}\right){〈\frac{\partial H_{\mathrm{int}}(q,Q)}{\partial Q_j}\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_i}〉}_{\mathrm{mc}}.\hfill \end{array}$$

(49)

Finally, with the definition (37) of the friction coefficient, we can write this result in the suggestive form

$$\langle \Delta P_i\Delta P_j\rangle =2k_{\mathrm{B}}T{\gamma }_{ij}\Delta t.$$

(50)

3.2.5. Evaluation of $\langle \Delta Q_i\Delta P_j\rangle$

Finally, we quickly verify that $\langle \Delta Q_i\Delta P_j\rangle$ has a quadratic (and higher) contribution in $\Delta t$. We start from (18e), plug in the expression (10) for $p_E\left({\mathrm{\Phi }}^{\prime }\right|\mathrm{\Phi })$ and use the formal solutions (3c), (3d) to assess the various contributions:

$$\begin{array}{cc}\hfill \langle \Delta Q_i\Delta P_j\rangle & =\int \mathrm{d}{\mathrm{\Phi }}^{\prime }(Q_i^{\prime }-Q_i)(P_j^{\prime }-P_j)\frac{\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\delta \left({\mathrm{\Phi }}^{\prime }-{\mathrm{\Phi }}_{\Delta t}(\varphi ,\mathrm{\Phi })\right)}{\mathrm{\Omega }\left(E\right)}\hfill \\ \hfill & =\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \left[{\left(Q_i\right)}_{\Delta t}(\varphi ,\mathrm{\Phi })-Q_i\right]\left[{\left(P_j\right)}_{\Delta t}(\varphi ,\mathrm{\Phi })-P_j\right]\delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\hfill \\ \hfill & =\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\left[{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\frac{{\left(P_i\right)}_{t^{\prime }}}{M}\right]\hfill \\ \hfill & \times \left[-{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\left(\frac{\partial H_{\mathrm{int}}(q_{t^{\prime }},Q_{t^{\prime }})}{\partial Q_j}+\frac{\partial H_{\mathrm{sys}}({\mathrm{\Phi }}_{t^{\prime }},{\lambda }_{t^{\prime }})}{\partial Q_j}\right)\right].\hfill \end{array}$$

(51)

Expanding the slowly varying system position ${\left(Q_i\right)}_{t^{\prime }}$ at lowest order $\Delta t$, we can approximate ${\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }\frac{{\left(P_i\right)}_{t^{\prime }}}{M}$ by $\frac{P_i}{M}\Delta t$ (cf. also (20)). We are then basically left with an average over $\Delta P_j$, for which we can re-use the results from Section 3.2.2. We thus obtain

$$\begin{array}{cc}\hfill \langle \Delta Q_i\Delta P_j\rangle & =\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\left[\frac{P_i}{M}\Delta t\right]\hfill \\ \hfill & \times [-{\int }_t^{t+\Delta t}\mathrm{d}{t}^{\prime }[\Delta t-(t^{\prime }-t)]\sum _k\frac{{\partial }^2{H}_{\mathrm{int}}(q_{t^{\prime }},Q)}{\partial Q_k\partial Q_j}\frac{P_k}{M}\hfill \\ \hfill & -\frac{\partial H_{\mathrm{int}}(q,Q)}{\partial Q_j}\Delta t-\frac{\partial H_{\mathrm{sys}}(\mathrm{\Phi },\lambda )}{\partial Q_j}\Delta t]+\mathcal{O}(\Delta t^2)\hfill \\ \hfill & =\mathcal{O}(\Delta t^2).\hfill \end{array}$$

(52)

3.3. Summary

Collecting all contributions that are linear in $\Delta t$, we have

$$\begin{array}{cc}\hfill \langle \Delta Q_i\rangle & =\frac{P_i}{M}\Delta t,\hfill \end{array}$$

(53a)

$$\begin{array}{cc}\hfill \langle \Delta P_i\rangle & =-\sum _j{\gamma }_{ij}\frac{P_j}{M}\Delta t+f_i(Q,\lambda )\Delta t,\hfill \end{array}$$

(53b)

$$\begin{array}{cc}\hfill \langle \Delta P_i\Delta P_j\rangle & =2k_{\mathrm{B}}T{\gamma }_{ij}\Delta t,\hfill \end{array}$$

(53c)

with the tensorial friction coefficient

$${\gamma }_{ij}=\frac{1}{k_{\mathrm{B}}T}{\int }_0^{\infty }\mathrm{d}t{〈\frac{\partial H_{\mathrm{int}}(q,Q)}{\partial Q_j}\frac{\partial H_{\mathrm{int}}(q_t,Q)}{\partial Q_i}〉}_{\mathrm{mc}}$$

(54)

All other averages or correlations are at least $\mathcal{O}(\Delta t^2)$.

It is straightforward to verify that a standard Langevin equation of the form

$$\begin{array}{cc}\hfill {\dot{Q}}_i\left(t\right)& =\frac{P_i\left(t\right)}{M},\hfill \end{array}$$

(55a)

$$\begin{array}{cc}\hfill {\dot{P}}_i\left(t\right)& =-\sum _j{\gamma }_{ij}\frac{P_j\left(t\right)}{M}+f_i(Q\left(t\right),\lambda \left(t\right))+\sqrt{2k_{\mathrm{B}}T}\sum _j{\left({\gamma }^{1/2}\right)}_{ij}{\xi }_j\left(t\right),\hfill \end{array}$$

(55b)

with mutually independent, unbiased, $\delta$-correlated white noise processes ${\xi }_j\left(t\right)$, generates identical average “displacements” and correlations in linear order $\Delta t$ (remember that the averages in (53) are conditioned on the initial configuration at the beginning of the time step $\Delta t$). Moreover, the diffusion coefficient quantifying the mean square displacement of the spatial coordinates Q is

$$D_{ij}=k_{\mathrm{B}}T{\left({\gamma }^{-1}\right)}_{ij},$$

(56)

obeying Einstein’s fluctuation–dissipation relation [2,3,4,5].

Strictly speaking, we did not demonstrate that the fluctuations behind the averages (53) are Gaussian. However, with the huge separation of time scales $\tau \ll \Delta t$, we can assume that there exists an “intermediate” time scale $\widetilde{\Delta t}$, with $\tau \ll \widetilde{\Delta t}\ll \Delta t$. It is still much larger than $\tau$, so all the above results apply to displacements over $\widetilde{\Delta t}$ as well, but it is considerably smaller than $\Delta t$, so displacements over $\Delta t$ are the result of many independent and identically distributed displacements over $\widetilde{\Delta t}$. According to the central limit theorem, the resulting dynamics on $\Delta t$ scales are then Gaussian-distributed with the moments and correlations given above in (53).

3.4. Rules of Calculus

Since we find that the statistical properties of the system momentum are generated by a white noise source, the question about the proper rules of (stochastic) calculus naturally arises, eminent in the procedure of calculating quantities like $〈P_i\Delta P_j〉$. We approach this question by considering averages of the form $〈g\left({\mathrm{\Phi }}^{\prime }\right)-g\left(\mathrm{\Phi }\right)〉$ in lowest order $\Delta t$, with an arbitrary (smooth) function g of the system state. In analogy to (10), the probability of observing a certain value $g\left({\mathrm{\Phi }}^{\prime }\right)$ is given as

$$p_E\left(g\left({\mathrm{\Phi }}^{\prime }\right)\right|\mathrm{\Phi })=\frac{\int \mathrm{d}\varphi \delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)\delta \left(g\left({\mathrm{\Phi }}^{\prime }\right)-g\left({\mathrm{\Phi }}_{\Delta t}(\varphi ,\mathrm{\Phi })\right)\right)}{\mathrm{\Omega }\left(E\right)},$$

(57)

such that we obtain

$$〈g\left({\mathrm{\Phi }}^{\prime }\right)-g\left(\mathrm{\Phi }\right)〉=\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \left[g\left({\mathrm{\Phi }}_{\Delta t}(\varphi ,\mathrm{\Phi })\right)-g\left(\mathrm{\Phi }\right)\right]\delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right).$$

(58)

Using our insights gained above, in Section 3.2, we expand this expression to lowest order in $\Delta t$,

$$\begin{array}{cc}\hfill 〈g\left({\mathrm{\Phi }}^{\prime }\right)-g\left(\mathrm{\Phi }\right)〉& =\frac{1}{\mathrm{\Omega }\left(E\right)}\int \mathrm{d}\varphi \left[\frac{\partial g\left(\mathrm{\Phi }\right)}{\partial \mathit{Q}}·\Delta \mathit{Q}+\frac{\partial g\left(\mathrm{\Phi }\right)}{\partial \mathit{P}}·\Delta \mathit{P}\right.\hfill \\ \hfill & \left.+\frac{1}{2}\sum _{i,j}\frac{{\partial }^{2}g\left(\mathrm{\Phi }\right)}{\partial P_i\partial P_j}\Delta P_i\Delta P_j\right]\delta \left(E-H_{\mathrm{bath}}(\varphi ,Q)\right)+\mathcal{O}(\Delta t^2)\hfill \\ \hfill & =\frac{\partial g\left(\mathrm{\Phi }\right)}{\partial \mathit{Q}}·\langle \Delta \mathit{Q}\rangle +\frac{\partial g\left(\mathrm{\Phi }\right)}{\partial \mathit{P}}·\langle \Delta \mathit{P}\rangle +\frac{1}{2}\sum _{i,j}\frac{{\partial }^{2}g\left(\mathrm{\Phi }\right)}{\partial P_i\partial P_j}\langle \Delta P_i\Delta P_j\rangle +\mathcal{O}(\Delta t^2)\hfill \\ \hfill & =\frac{\partial g\left(\mathrm{\Phi }\right)}{\partial \mathit{Q}}·\langle \Delta \mathit{Q}\rangle +\frac{\partial g\left(\mathrm{\Phi }\right)}{\partial \mathit{P}}·\langle \Delta \mathit{P}\rangle +k_{\mathrm{B}}T\sum _{i,j}\frac{{\partial }^{2}g\left(\mathrm{\Phi }\right)}{\partial P_i\partial P_j}{\gamma }_{ij}\Delta t+\mathcal{O}(\Delta t^2),\hfill \end{array}$$

(59)

where we use (50) in the last step. With the standard rules of (“non-stochastic”) calculus, the expansion in lowest order would result only in the first two terms. The third term corresponds exactly to the additional contribution characteristic for the so-called Stratonovich rule of stochastic calculus [3,4]. Denoting this rule by ∘, we can therefore write

$$〈g\left({\mathrm{\Phi }}^{\prime }\right)-g\left(\mathrm{\Phi }\right)〉=\langle \Delta g\left(\mathrm{\Phi }\right)\rangle =〈\frac{\partial g\left(\mathrm{\Phi }\right)}{\partial \mathrm{\Phi }}\circ \Delta \mathrm{\Phi }〉.$$

(60)

As a concrete example, we consider $g\left(\mathrm{\Phi }\right)={\mathit{P}}^2$:

$$\langle \Delta \left({\mathit{P}}^2\right)\rangle =2\langle \mathit{P}\circ \Delta \mathit{P}\rangle =2\mathit{P}·\langle \Delta \mathit{P}\rangle +2k_{\mathrm{B}}TTr\left(\gamma \right)\Delta t.$$

(61)

3.5. Connection to the Fokker-Planck Equation

In Section 3.1, we prove that the effective system dynamics are Markovian on $\Delta t$ time scales; see Equation (12). Therefore, the dynamics also obey the Chapman–Kolmogorov equation [2,3,5], which, by the standard Kramers–Moyal expansion [2,3,5], is recast into the (forward) Kolmogorov equation (adjusted to our notation here),

$$\begin{array}{c}\frac{\partial }{\partial t^{\prime }}p({\mathrm{\Phi }}^{\prime },t^{\prime }|\mathrm{\Phi },t)=-\sum _i\frac{\partial }{\partial P_i^{\prime }}{A}_i({\mathrm{\Phi }}^{\prime },t^{\prime })p({\mathrm{\Phi }}^{\prime },t^{\prime }|\mathrm{\Phi },t)-\sum _i\frac{\partial }{\partial Q_i^{\prime }}{B}_i({\mathrm{\Phi }}^{\prime },t^{\prime })p({\mathrm{\Phi }}^{\prime },t^{\prime }|\mathrm{\Phi },t)\hfill \\ \hfill +\frac{1}{2}\sum _{i,j}\frac{{\partial }^2}{\partial P_i^{\prime }\partial P_j^{\prime }}{A}_{ij}({\mathrm{\Phi }}^{\prime },t^{\prime })p({\mathrm{\Phi }}^{\prime },t|\mathrm{\Phi },t)+\frac{1}{2}\sum _{i,j}\frac{{\partial }^2}{\partial Q_i^{\prime }\partial Q_j^{\prime }}{B}_{ij}({\mathrm{\Phi }}^{\prime },t^{\prime })p({\mathrm{\Phi }}^{\prime },t^{\prime }|\mathrm{\Phi },t),\end{array}$$

(62a)

with the coefficients (using $t^{\prime }-t=\Delta t$ as before)

$$\begin{array}{cc}\hfill A_i(\mathrm{\Phi },t)& =\underset{\Delta t\to 0}{lim}\frac{1}{\Delta t}\int \mathrm{d}{\mathrm{\Phi }}^{\prime }p({\mathrm{\Phi }}^{\prime },t^{\prime }|\mathrm{\Phi },t)(P_i^{\prime }-P_i),\hfill \end{array}$$

(62b)

$$\begin{array}{cc}\hfill B_i(\mathrm{\Phi },t)& =\underset{\Delta t\to 0}{lim}\frac{1}{\Delta t}\int \mathrm{d}{\mathrm{\Phi }}^{\prime }p({\mathrm{\Phi }}^{\prime },t^{\prime }|\mathrm{\Phi },t)(Q_i^{\prime }-Q_i),\hfill \end{array}$$

(62c)

$$\begin{array}{cc}\hfill A_{ij}(\mathrm{\Phi },t)& =\underset{\Delta t\to 0}{lim}\frac{1}{\Delta t}\int \mathrm{d}{\mathrm{\Phi }}^{\prime }p({\mathrm{\Phi }}^{\prime },t^{\prime }|\mathrm{\Phi },t)(P_i^{\prime }-P_i)(P_j^{\prime }-P_j),\hfill \end{array}$$

(62d)

$$\begin{array}{cc}\hfill B_{ij}(\mathrm{\Phi },t)& =\underset{\Delta t\to 0}{lim}\frac{1}{\Delta t}\int \mathrm{d}{\mathrm{\Phi }}^{\prime }p({\mathrm{\Phi }}^{\prime },t^{\prime }|\mathrm{\Phi },t)(Q_i^{\prime }-Q_i)(Q_j^{\prime }-Q_j).\hfill \end{array}$$

(62e)

Here, $p({\mathrm{\Phi }}^{\prime },t^{\prime }|\mathrm{\Phi },t)$ is the transition probability density from Equation (10) (including explicitly the time arguments, but skipping the subscript E). The coefficients (62) are given by exactly the moments we already calculated, cf. Equation (18). From the results listed in Equation (53) (and by remembering that all other moments are $\mathcal{O}(\Delta t^2)$), we immediately obtain

$$\begin{array}{cc}\hfill A_i(\mathrm{\Phi },t)& =-\sum _j{\gamma }_{ij}\frac{P_j}{M}+f_i(Q,\lambda ),\hfill \end{array}$$

(63a)

$$\begin{array}{cc}\hfill B_i(\mathrm{\Phi },t)& =\frac{P_i}{M},\hfill \end{array}$$

(63b)

$$\begin{array}{cc}\hfill A_{ij}(\mathrm{\Phi },t)& =2k_{\mathrm{B}}T{\gamma }_{ij},\hfill \end{array}$$

(63c)

$$\begin{array}{cc}\hfill B_{ij}(\mathrm{\Phi },t)& =0.\hfill \end{array}$$

(63d)

The forward Kolmogorov Equation (62a) thus reduces to the standard Fokker–Planck equation for underdamped Brownian motion [2,3,4,5].

4. Discussion and Conclusions

Starting from first principles with a full microscopic description of a huge thermal bath (“environment”), interacting with a small system of interest, we demonstrate that the effective equation of motion for the system degrees of freedom alone is given by the standard Langevin Equation (55). To the best of our knowledge, the approach presented here is new. It is based on two assumptions: (i) the bath is in thermal equilibrium at all times, and (ii) there is a vast separation of time scales between the bath and system, i.e., the system degrees of freedom are slow coordinates that evolve on time scales much larger than the time scales characterizing the bath dynamics (e.g., molecular collision times). These are the standard conditions for which the Langevin equation is known to be valid as an effective description of the system dynamics. Accordingly, they sooner or later enter other known derivations of the Langevin equation, like the famous projection operator technique (see, for instance, [1,2,10,11,12,13,14]), or the intriguing procedure presented in Section 3.4.1 of Reference [15].

Here, we take these assumptions as a starting point and, in that way, are able to assess the statistical properties of the system dynamics without directly projecting or integrating out [16] the fast bath degrees of freedom. Moreover, the microscopic origin of the various contributions in the Langevin equation becomes evident. In particular, the friction coefficient is identified with the correlations of the interaction forces (see Equation (39)), and the fluctuation–dissipation relation (56) is traced back to the microcanonical definition of temperature (9). Moreover, the pertinent rules of stochastic calculus are identified by our approach to follow the Stratonovich prescription (see Section 3.4). Our derivation can be straightforwardly generalized to many Brownian particles interacting via pairwise interaction potentials as long as “hydrodynamic” interactions mediated by the thermal bath are neglected.

It will be interesting to explore how our approach generalizes to other situations (e.g., the presence of magnetic fields), to other configuration spaces (e.g., rotational Brownian motion [2], or, more generally, Brownian motion on manifolds) or to other system–bath complexes that can be described by Hamilton’s equations of motion (e.g., classical spin systems [17]).