**1. Introduction**

The thermodynamics and statistical physics of particles at equilibrium is a standard part of the undergraduate curriculum. The First and Second Law of Thermodynamics are powerful concepts that lead the way to the explanation of many real-life phenomena. Further development led to notions such as the Boltzmann Distribution, the Fluctuation-Dissipation Theorem, Onsager's Reciprocal Relation, and Microscopic Reversibility [1]. Even setups that are close-to-equilibrium can often be successfully analyzed with these ideas. No general theory, however, is available for systems that are far-from-equilibrium. None of the above laws and notions apply in that case.

Imagine a liquid in which "active" particles are suspended. Such "active" particles can be bacteria that propel themselves, i.e., swim. These can also be particles that are manipulated through fields from the outside. Obviously, energy is pumped into such systems and no First Law or any of the concepts mentioned in the previous paragraph applies. Over the last two decades, setups with active particles have been the subject of much experimental and theoretical research.

There are many different ways to model the movements of active particles. One can, for instance, assume that the particle has the same speed all the time and that the change of the direction of motion follows a diffusion equation [2]. The "Run-and-Tumble" model is a more discrete version of this and it is inspired by the way that *Escherichia coli* bacteria

**Citation:** Yuvan, S.; Bier, M. Accumulation of Particles and Formation of a Dissipative Structure in a Nonequilibrium Bath. *Entropy* **2022**, *24*, 189. https://doi.org/ 10.3390/e24020189

Academic Editor: Antonio M. Scarfone

Received: 17 January 2022 Accepted: 25 January 2022 Published: 27 January 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

move [3]. Here the particle or bacteria covers a finite-length straight segment at a constant speed. After coming to a stop, it lingers for a moment. It "tumbles" and then picks a new random direction for the next run. There are also different ways to let the active particle interact with the wall of the reservoir in which it swims.

In our analysis below, we focus on the 2D random walk: At every timestep, a direction is picked randomly and a displacement is drawn from a zero-centered distribution (cf. Figure 1). We let the random walks happen in a confinement. Whenever the particle hits the wall, it comes to a standstill. Subsequently, it only moves away from the wall again if a random displacement makes it move inside the circular confinement.

If displacements are drawn from a zero-average Gaussian distribution, we eventually see a homogeneous distribution of particle positions over the entire domain. However, if we instead draw distances from a so-called *α*-stable distribution (sometimes called a Lévystable distribution) [4–7], a nonhomogeneous distribution develops.

The Gaussian distribution has an exponential tail, i.e., *p*2(*ξ*) ∝ exp −*ξ*2/2*σ*<sup>2</sup> as *ξ* → ±∞. Here *σ* denotes the standard deviation of the Gaussian. The rapid convergence to zero of the exponential tail means that the probability to make a big jump is very small and effectively negligible. Figure 1a shows this clearly.

For an *α*-stable distribution, the asymptotic behavior is described by a power law:

$$p\_{\mathfrak{a}}(\mathfrak{z}) \propto |\mathfrak{z}|^{-\mathfrak{a}-1} \quad \text{as} \ \mathfrak{z} \to \pm \infty. \tag{1}$$

Here *α* is the so-called stability index for which we have 0 < *α* < 2. For *α* = 2, the Gaussian is re-obtained. The power law converges slower than the exponential. A result of this is that outliers, i.e., large "Lévy jumps", regularly occur (see Figure 1b). Ultimately, the Lévy walk resembles a run-and-tumble walk, but, following Equation (1), the Lévy jumps have no characteristic length and the average length of a Lévy jump actually diverges.

**Figure 1.** Random walk in a circular domain. Whenever the particle hits the wall, it comes to a standstill and later only moves again when a computed step leads to a movement inside the circle. For every step, the direction is picked randomly and the displacement is drawn from a (**a**) Gaussian distribution or from a (**b**) Lévy-stable distribution. The circle has a radius of 20. Both distributions are symmetric around zero. The Gaussian distribution has a standard deviation of <sup>√</sup>2. For the Lévy-stable distribution, we have *α* = 1 and a scale factor of *σ* = 1.

The Central Limit Theorem [8] tells us that the Gaussian distribution is what ensues when an outcome is the result of multiple stochastic inputs. However, the theorem only applies if all of the constituent stochastic inputs have a *finite* standard deviation. For stochastic inputs with infinite standard deviations, the *α*-stable distribution is what results.

The *α*-stable distribution is a standard feature of the *Mathematica* software package and the programming for a simulation as the one leading to Figure 1b is a matter of just a few lines of code. The probability density of the *α*-stable distribution is given by a big and cumbersome formula [9] and we will not elaborate on it.

Alpha-stable distributions do not just provide a good model for the behavior of active particles. It turns out that power-law tails commonly occur in systems that are far-fromequilibrium with no active particles involved. Almost 60 years ago, Benoit Mandelbrot discovered that variations in the price of cotton futures follow a distribution with an *α* = 1.7 power-law tail [10,11]. More power-law tails and *α*-stable distributions were identified in the 1990s [12–14] when desktop computers became available that could rapidly and easily perform the necessary data processing. As of yet, there is no complete and general theory to explain how and why *α*-stable distributions are connected to far-from-equilibrium. In this sense, the *α*-stable distributions are like 1/ *f*-noise [11,15]. The connection of farfrom-equilibrium with *α*-stable distributions and 1/ *f*-noise is still for the most part, a phenomenological one.

Nevertheless, as mentioned above, nonequilibrium characteristics do emerge when, instead of Gaussian noise, Lévy noise is added to particle dynamics. Take a particle doing Brownian motion on a potential *V*(*x*). Microscopic reversibility means that every trajectory *x*(*t*) from an initial position *xi* to a final position *xf* and taking a time Δ*t*, is traversed equally often in forward and backward direction. Microscopic reversibility is an equilibrium feature that is implied by the fact that there can be no arrow of time for a system at equilibrium, i.e., there must be time-reversal symmetry. In 1953, Onsager and Machlup gave mathematical rigor to this idea when they proved that with Gaussian noise, the most likely trajectory up a potential barrier is the reverse of the most likely trajectory down that same barrier [16–18]. It can also be rigorously proven that for Lévy noise, the most likely trajectory up a potential barrier is not the reverse of the most likely trajectory down a potential slope [19]. The presence of Lévy noise breaks the time-reversal symmetry that is implicit in equilibrium [20].

For the setup that is depicted in Figure 1b, the violation of time-reversal symmetry is in the interaction of the particle with the wall. Elastic collisions have time-reversal symmetry and had we taken the particle in Figure 1b to collide elastically with the wall, forward and backward trajectories would have been indistinguishable. Lévy jumps are rare, but because of their length, they are likely to end at the wall. Once the particle is located at the wall, the probability that the first subsequent step is already a Lévy jump away from the wall is small. Moreover, only a step that leads to a movement inside the reservoir will be processed in the simulation. Thus, the particle can "linger" near the wall after hitting it. In the end it appears as if it is easier to get to the wall than it is to get away from it, i.e., it looks as if there is reduced mobility near the wall. Figure 2 shows how this is the case on a 1D interval.

In the previous paragraph, we put the finger on something that applies generally for active particles in a confinement. They do not distribute homogeneously, but instead accumulate near a wall. It furthermore appears that the accumulation is stronger if the wall has a stronger inward curvature [21]. Active particles tend to get stuck in nooks and corners of a confinement and even more so if the nooks and corners are tighter. This is the phenomenon that we will elaborate on below.

The way Lévy particles distribute on a confined 1D segment (cf. Figure 2) can be described with a Fractional Fokker–Planck Equation [22]. The steady-state solution of that equation is available [23]. We show in Appendix A how this solution readily generalizes to higher dimensional setups. Below we examine how Lévy particles distribute over two connected reservoirs where one reservoir is a scaled down version of the other. We will see a deviation from the homogeneous distribution that is obtained when the noise is Gaussian and when equilibrium theory applies.

Suppose we have a volume *V* with *N* indistinguishable particles in it. We partition the initially empty *V* into two reservoirs of a volume *V*/2 each. Next the particles are inserted. Each reservoir has a probability of 1/2 to receive each particle. Eventually, the probability for all particles to end up in one particular reservoir is 2−*N*. The probability for an equal

distribution over the two reservoirs is ( *<sup>N</sup> <sup>N</sup>*/2)2−*N*. The binomial coefficient ( *<sup>N</sup> <sup>N</sup>*/2) grows very rapidly with *N*.

**Figure 2.** A Lévy walk on the interval −1 ≤ *x* ≤ 1 (cf. Equation (3)). The value of the stability index is *α* = 0.8. Whenever the particle hits *x* = ±1, it stays there until an iteration occurs in the direction away from the wall. The red curve shows the analytic solution (cf. Equation (4)). The normalized histogram is the result of a numerical simulation of Equation (3); the timestep was Δ*t* = 0.001, there were 107 iterations, and the scale factor of the symmetric, zero-centered Lévy distribution was taken to be one.

The reason that the air in a room never spontaneously concentrates in one half of the room is that there is just one way to put all molecules in one chosen half and ( *<sup>N</sup> <sup>N</sup>*/2) ways to distribute them equally. In other words, the macrostate in which all air is concentrated in one particular half of the room has *one microstate* and the macrostate with a homogeneous air distribution over the entire room has ( *<sup>N</sup> <sup>N</sup>*/2) *microstates*. The entropy of a macrostate can be defined as a scalar value that is proportional to the logarithm of the number of microstates of that macrostate [1]. In this case, it is obvious that the homogeneous distribution leads to maximal entropy.

With a partition and a pump it is, of course, possible to bring all of the air molecules to one half of the room. Such a process requires energy and with standard thermodynamics, the involved energies can be calculated. That energy-consuming, active particles can accumulate in a smaller subvolume does not violate laws of nature, and it is also possible to calculate the entropy change associated with such accumulation. We will perform such a calculation.

The ultimate goal would be a Lévy-noise-equivalent of entropy. This would be a quantity that takes its extreme value when Lévy-noise-subjected particles reach a steady state distribution. The Kullback–Leibler divergence [24] is a positive scalar value that can be thought of as a "distance" between two given distributions. The Kullback–Leibler divergence between the steady state distribution and another distribution could be a good candidate. With tools like Noether's Theorem, alternative formulations of activeparticle statistical mechanics and of the Fractional Fokker–Planck Equation have been derived [25,26], with work in this direction appearing to be promising.

No general formalism is developed in this article, but we present a setup where the entropy decrease associated with the accumulation can be readily described with simple and intuitive formulae. The nonhomogeneous steady-state distributions that develop in the presence of nonequilibrium noise can be interpreted as dissipative structures [27]. The deviation from homogeneity decreases the entropy. However, active particles pump energy into the system and the dissipative structure ultimately facilitates a steady-state dissipation of energy and production of entropy.

## **2. The 1D and 2D Random Walk in a Confined Domain**

For a particle in 1D, Brownian motion is commonly described with a Langevin equation:

$$
\dot{\mathfrak{x}}(t) = \sqrt{2D} \,\, \mathfrak{z}\_2(t), \tag{2}
$$

where *x*(*t*) is the time-dependent position of the particle, • ≡ *d*/*dt*, *D* is the diffusion coefficient, and *ξ*2(*t*) is a stochastic function that describes the effect of collisions with molecules in the medium. To account for the effect of such collisions, a random number *θ*2,*<sup>i</sup>* is drawn at the *i*-th timestep. In a simulation with finite timesteps of Δ*t*, we then take <sup>Δ</sup>*xi* <sup>=</sup> <sup>√</sup>2*<sup>D</sup> <sup>ξ</sup>*2,*i*(*ti*) <sup>Δ</sup>*<sup>t</sup>* <sup>=</sup> <sup>√</sup>2*D*Δ*<sup>t</sup> <sup>θ</sup>*2,*i*. If <sup>Δ</sup>*<sup>t</sup>* is large enough to contain a significant number of collisions, then the aforementioned Central Limit Theorem [8] can be invoked to justify drawing the *<sup>θ</sup>i*'s from a zero-average Gaussian distribution. Upon taking *θ*<sup>2</sup> 2,*i* = 1, we readily come to the traditional equation for the average squared distance that is diffused in a time interval <sup>Δ</sup>*t*: Δ*x*2 = <sup>2</sup>*D*Δ*t*.

The equation Δ*x*2 = <sup>2</sup>*D*Δ*<sup>t</sup>* does not contain a characteristic timescale. It is in order for the scale-free diffusion equation to ensue that we need to "adjust" the *θ*2,*i*'s and take *ξ*2(*ti*) = *θ*2,*i*/ <sup>√</sup>Δ*<sup>t</sup>* in Equation (2).

In case of the 1D Lévy flight, we have for the stochastic ordinary-differential-equation and its discretized version, respectively:

$$
\dot{\mathfrak{x}} = \sigma \,\, \mathfrak{J}\_a(t) \quad \text{and} \quad \Delta \mathfrak{x}\_{\dot{i}} = \sigma \,\, \theta\_{\mathfrak{a},\dot{i}} \,\, \Delta t^{1/a} \,. \tag{3}
$$

Now the values for *θα* are drawn from a symmetric, zero-centered *α*-stable distribution with a value of one for its scale factor. The Lévy flight is still scale-free, but because *θ*<sup>2</sup> *α*,*i* → ∞ for 0 < *α* < 2, there is no longer a traditional diffusion equation and *σ* is a mere scale factor.

Figure 1 shows simulations of 2D random walks. At every timestep, a direction is chosen randomly from a flat distribution between zero and 2*π*. The displacement is the result of a random draw from a Gaussian distribution (Figure 1a) or from an *α*-stable distribution (Figure 1b). Both the Gaussian walk and Lévy walk are isotropic, i.e., taken from the center of the circle, all directions are equivalent. A generalization to more than 2 dimensions is readily formulated and simulated. The random walks then occur inside a ball with a finite radius. Whenever the domain boundary is hit, the particle comes to a standstill. For *α* = 2, the random walk is symmetric under time reversal. However, as was already mentioned in the Introduction, for 0 < *α* < 2 the time-reversal symmetry is broken. It is not hard to understand why this is the case. When the particle is followed in forward time, we will often see a Lévy jump that makes the particle come to a standstill at the domain boundary. More rare will be a large jump from the domain boundary into the interior. When a movie of the moving particle is played backward, it will be the other way round. The forward and backward played movie are distinguishable.

Figure 2 shows the position distribution that results after a many-step 1D simulation on −1 ≤ *x* ≤ 1 for *α* = 0.8. For *α* = 2, a flat distribution results. However for 0 < *α* < 2, the Lévy jumps that terminate at *x* = ±1 and the decreased mobility there lead to an increased probability density near *x* = ±1. The Langevin Equation, Equation (3), can be equivalently formulated as a fractional Fokker–Planck Equation for the evolution of a probability distribution, i.e., *∂<sup>t</sup> p*(*x*, *t*) = *σα∂α <sup>x</sup> p*(*x*, *t*). The stationary distribution is then obtained as the solution of the ordinary differential equation that results when the left hand side is set equal to zero. The fractional derivatives are nontrivial, but in Ref. [23], a solution for the 1D case is presented:

$$p\_{st}(\mathbf{x}) = \frac{2^{1-a}\Gamma(a)}{\Gamma^2(a/2)} \left(1 - \mathbf{x}^2\right)^{a/2 - 1},\tag{4}$$

where Γ(.) denotes the gamma function. Figure 2 shows this solution together with the results of the Langevin simulation. In Appendix A, we show with symmetry arguments that the normalized (<sup>1</sup> <sup>−</sup> *<sup>r</sup>*2)*α*/2−1-form generalizes to the *<sup>n</sup>*D case, with *<sup>r</sup>* being the distance from the center of the ball.

It is worth noting that the U-shaped function as in Equation (4) and Figure 2 has been encountered in other systems in stochastic dynamics. For *α* = 1, the Lévy stable distribution is actually the Cauchy distribution, *p*1(*ξ*)=(1/*π*)(*σ*/(*σ*<sup>2</sup> + *ξ*2)). For *α* = 1 and upon taking *<sup>x</sup>* = <sup>2</sup>*<sup>u</sup>* − 1, Equation (4) turns into *<sup>p</sup>*(*u*)=(1/*π*)(*u*(<sup>1</sup> − *<sup>u</sup>*))−1/2, where 0 ≤ *<sup>u</sup>* ≤ 1. This is called the arcsine distribution because the cumulative distribution yields an arcsine: *u* <sup>0</sup> *p*(*u* ) *du* = (2/*π*) arcsin <sup>√</sup>*u*. In 1939, it was the same Paul Lévy who derived that the arcsine distribution emerges in the following case [28]. Let a 1D Brownian walk of duration *t* start at *x* = 0. Next look at the fraction of time that the Brownian particle spends on the positive semi-axis. It is found that these fractions follow an arcsine distribution. This is called the arcsine law. Recently it has been discovered that arcsine laws occur more generally [29]. Driven mesoscopic systems are obviously out-of-equilibrium, but also in such systems, an arcsine law results when one considers, for instance, fractions of time that a current stays above its average value. Arcsine laws in nonequilibrium setups is currently a much researched topic [30,31] and Equation (4) may be a manifestation of something deeper and more general.

#### **3. Two Connected Semicircular Reservoirs**

Imagine a semicircular 2D domain with radius *R*<sup>1</sup> as in Figure 3. There is a small opening with a width *d* that gives access to a semicircular domain with radius *R*2. We have *R*<sup>2</sup> < *R*1. Next imagine a large number of particles in this system. The particles are subjected to Lévy noise. In Appendix B, it is derived how there is a net flow into the smaller reservoir if both reservoirs have the same homogeneous particle density. Thus, when starting from thermodynamic equilibrium, a higher density develops in the smaller reservoir once Lévy noise starts being applied.

**Figure 3.** Two semicircular reservoirs with a small opening between them. The system contains a large number of noisy particles. At each timestep, each particle moves in an arbitrary direction with a displacement that is drawn from a Gaussian distribution or a Lévy-stable distribution as in Figure 1a,b. If a particle hits a semicircular wall, it comes to a standstill and only moves again if a computed displacement leads to motion inside the system. If a particle hits the straight vertical wall, it bounces elastically. For Gaussian noise, the system goes to an equilibrium with equal concentration on both sides of the opening. However, when the particles are subjected to Lévy noise, the steady state has an accumulation in the smaller reservoir.
