*3.2. Entropies and Energies Associated with Lévy Noise*

The nonhomogeneous distributions shown in Figures 2 and 5 essentially function as dissipative structures [27]. The depicted nonhomogeneous steady-state distributions represent a lower entropy than homogeneous distributions. However, these lower-entropy structures facilitate the transfer and dissipation of energy at steady state. The transferred energy comes in through the non-thermal motion of the active particles. It is next dissipated and released. Ultimately the low-entropy dissipative structures help the energy throughput and the entropy production.

As a result of the divergent standard deviation of the *α*-stable noise, the energy that is dissipated per unit of time is in principle infinite. The finite container size, however, truncates the Lévy jumps and make the aforementioned standard deviation of the jump sizes finite. We will not elaborate on this. What we will instead focus on in this subsection is the entropy decrease that is associated with the apparent nonhomogeneous distribution shown in Figure 5.

Imagine that the steady flow of energy that maintains the dissipative structure is suddenly halted. Such halting is straightforward if the active-particle-motion is, for instance, driven by magnetic forces or by optics. The distribution in Figure 5 will then homogenize. Such homogenization implies an increase in entropy and a concurrent decrease in free energy. Below we will find remarkably concise analytic expressions for the entropy change.

The relaxation towards homogeneity is two-part. First there is an intra-reservoir relaxation inside each of the two reservoirs to a spatially homogeneous spread. Next there is the slower relaxation between the two reservoirs towards a ratio *ϕ*1/*ϕ*<sup>2</sup> = *V*1/*V*<sup>2</sup> = *R*<sup>2</sup> 1/*R*<sup>2</sup> 2.

The entropy change associated with the intra-reservoir relaxations is hard to compute for the semicircular reservoirs of Figures 3 and 5. However, for a circular reservoir as in Figure 1, it is easier and no resort to numerics is necessary. We take *pini*(*r*)=(*α*/2*π*)(<sup>1</sup> <sup>−</sup> *<sup>r</sup>*2)*α*/2−<sup>1</sup> as the initial distribution and *pfin*(*r*) = 1/*π* as the final homogeneous distribution. It is well known that for a discrete set of probabilities, *pi*, the associated entropy is given by *S* = −Σ*<sup>i</sup> pi* log *pi*. However, this summation cannot be straightforwardly extended to an integral for the case of a continuous probability density *p*(*r*). An obvious issue is that density is not dimensionless and that a logarithm can only be taken of a dimensionless quantity. In Ref. [33], it is explained how a sensible definition is only obtained after introducing another probability density that functions as a measure. We then obtain what is known as the relative entropy or Kullback–Leibler divergence [24]:

$$D\_{\rm KL}(p\_{fin}||p\_{ini}) = \int\_{r \le 1} p\_{fin}(r) \log \left(\frac{p\_{fin}(r)}{p\_{ini}(r)}\right) dr. \tag{16}$$

When working out this integral, it is important to realize that the integration is from *r* = 0 to *r* = 1 over the area of a circle and that a term 2*πr* needs also be included. With the above expressions for *pini*(*r*) and *pfin*(*r*), we find after some algebra that *D*KL(*pfin*||*pini*) = −1 + *α*/2 + log(*α*/2). No such easy analytic solution ensues for more than two dimensions or even in the 1D case. The Kullback–Leibler divergence can be thought of as a kind of distance between two probability densities. However, it is generally not symmetric in the two involved distributions. In our case, we find *D*KL(*pini*||*pfin*) = −1 + 2/*α* + log(2/*α*). Both *D*KL(*pfin*||*pini*) and *D*KL(*pini*||*pfin*) are remarkably simple expressions; they are continuous and concave up as *α* increases and reduce to zero for *α* = 2.

The speed of the inter-reservoir relaxation depends on the size of the opening. For the small opening that is necessary for our approximations to be accurate, it will generally be slower than the intra-reservoir relaxation. For the inter-reservoir relaxation, the basic quantity is the probability to be in either of the two reservoirs. We go back to the basics to calculate what the entropy is for a given distribution over the two reservoirs.

In the Statistical Physics context, entropy is commonly defined as proportional to the logarithm of the number of microstates [1]. Imagine that there are *N* identical particles in the setup of Figures 3 and 5. Here *N* is taken to be very large. In case of equilibrium, the number of particles in a reservoir is proportional to the volume *Vi* = *πR*<sup>2</sup> *<sup>i</sup>* /2 of a reservoir. With *ϕiN* identical particles in reservoir *i*, the number of microstates in each of the two reservoirs is given by:

$$
\Omega\_i = \frac{V\_i^{\varphi\_i N}}{(\varphi\_i N)!}.\tag{17}
$$

The numerator has the *ϕiN*-exponent because it is for each particle that the number of microstates is proportional to the volume. The microstate is the same, however, when two or more particles are exchanged. The denominator takes this into account and denotes the number of permutations among *ϕiN* particles. With the definition *S* = log Ω and using Stirling's approximation [1] (log *N*! = *N* log *N*, if *N* is very large), we derive:

$$S\_{\bar{i}} = \varphi\_{\bar{i}} N \log \left( \frac{V\_{\bar{i}}}{\varphi\_{\bar{i}} N} \right),\tag{18}$$

where "log" denotes the natural logarithm. As was mentioned before, at thermodynamic equilibrium the fraction of particles in a reservoir is proportional to the volume of that reservoir, i.e., *ϕ<sup>i</sup>* ∝ *Vi*. The argument of the logarithm in Equation (18) is then the same constant

for both reservoirs. This leads to *Si* ∝ *ϕi*, as should be expected from an equilibriumthermodynamics perspective.

We take for the total volume and the total entropy *Vtot* = *V*<sup>1</sup> +*V*<sup>2</sup> and *Stot* = *S*<sup>1</sup> + *S*2, respectively. It is next derived from Equation (18) that *Stot* = *N*(*ϕ*<sup>1</sup> log(*V*1/*ϕ*1)+ *ϕ*<sup>2</sup> log(*V*2/*ϕ*2)) − *N* log *N*. The additive *N* log *N*-term is the same for all values of *α*. As it is only differences in entropy that matter, we discard this term. For the entropy per particle, *stot* = *Stot*/*N*, it is next found:

$$s\_{tot} = \varphi\_1 \log\left(\frac{V\_1}{\varphi\_1}\right) + \varphi\_2 \log\left(\frac{V\_2}{\varphi\_2}\right). \tag{19}$$

Figure 6 depicts *stot* as a function of *α* following Equation (19). We took *Vtot* = 1 (leading to *V*<sup>1</sup> = *R*<sup>2</sup> 1/(*R*<sup>2</sup> <sup>1</sup> + *<sup>R</sup>*<sup>2</sup> <sup>2</sup>) and *<sup>V</sup>*<sup>2</sup> = *<sup>R</sup>*<sup>2</sup> 2/(*R*<sup>2</sup> <sup>1</sup> + *<sup>R</sup>*<sup>2</sup> <sup>2</sup>)) and *R*<sup>1</sup> = 10*R*2. For the dashed curve, Equation (13) was used to come to the values of *ϕ*<sup>1</sup> and *ϕ*2. For the solid curve the improved approximation, Equation (15) was used with *r*<sup>0</sup> = 0.05. The curves appear almost indistinguishably close. It is important to realize that this entropy also represents free energy. The free energy release associated with the equilibration can be obtained by multiplying the entropy (cf. Equation (19)) with the temperature. Again we emphasize that Equation (19) is related to just the inter-reservoir relaxation and does not incorporate intra-reservoir relaxation.

**Figure 6.** Given the setup of Figures 3 with *Vtot* = 1 and *R*<sup>1</sup> = 10*R*2, the curves show the entropy per particle, *stot*, as a function of the stability parameter *α* of the Lévy noise. The nonequilibrium noise leads to a concentration difference between the two reservoirs. The associated entropy decrease *stot* is obtained by substituting into Equation (19) the approximate ratio according to Equation (13) (dashed curve) and according to Equation (15) (solid curve). For Equation (15) we took *r*<sup>0</sup> = 0.05, i.e., the value that led to good agreement with the stochastic stimulation (cf. Figure 4).

There is more thermodynamic way to derive the right-hand side of Equation (19) as the energy per particle that is invested in the building of the dissipative structure. With intrareservoir equilibrium established, the chemical potential *μ* that is driving flux through the opening is the logarithm of the concentration ratio [1]. If we let *φ* be the fraction of the particles in the smaller reservoir, then we have *μ*(*φ*) = log *<sup>φ</sup> V*2 <sup>−</sup> log <sup>1</sup>−*<sup>φ</sup> V*1 . The energy that is dissipated when an infinitesimal fraction *dφ* follows the potential and flows through the opening is *μ*(*φ*) *dφ*. The entire equilibration takes *φ* from *ϕ*<sup>2</sup> to *V*2. After some algebra and setting the temperature and the Boltzmann constant all equal to unity, it is found that the resulting total-equilibration-energy integral reduces to −*stot* (cf. Equation (19)).

Equations (13) and (19) are concise and intuitive. Equation (13) is already a fairly accurate approximation. Given the geometry of the system and the value of *α*, Equation (13) gives the distribution over the two reservoirs. Equation (19) tells us what entropy decrease and what free energy "investment" is associated with the concentration difference between the reservoirs that gets established due to the active particle movement. It gives a measure for how far the system is driven from equilibrium by the active particle motion.

#### **4. Discussion**

In this article we explored a significant consequence of a bath in which particles velocities are Lévy-stable distributed. With the ordinary Gaussian velocity distribution that is associated with equilibrium systems, the maximization of entropy leads to particles homogeneously distributing in the confined domain. With a Lévy-stable distribution for the velocities, larger concentrations occur near the walls and in the smaller cavities. We have analytic expressions for the distribution of Lévy particles in the circular and the spherical domain. For the two connected reservoirs as depicted in Figure 3, we have derived a good approximation for the concentration difference between the reservoirs at steady state. We have presented an accounting of the entropies, and ensuing energies, for such divergence from equilibrium.

We have interpreted the nonhomogeneous particle distribution (cf. Figure 5) as a dissipative structure, i.e., a lower-entropy arrangement of particles that facilitates a larger dissipation of energy and concurrent larger production of entropy. There is nothing about heat conduction in the equations. However, it is tempting to hypothesize that with the particles being closer to the surface area, the system would be better able to transfer heat to the environment and do so at a larger rate.

In the 1990s, experiments were performed in which DNA, RNA, and proteins were manipulated on the molecular scale. This commonly involved breaking of molecular bonds. The involved energies were significantly larger than the *kBT* that can be considered as the quantum of thermal energy. In such far-from-equilibrium processes, Onsager's reciprocal relations and other close-to-equilibrium concepts are no longer valid. Fortunately, at about the same time, theory was developed to handle fluctuations in far-from-equilibrium conditions. The Fluctuation Theorem [34] and the Jarzynski Equation [35,36] could very accurately account for the results of experiments in which microscopic beads were pulled by optical tweezers [37] and experiments in which RNA was forcibly unfolded [38,39]. However, it should be realized that the Fluctuation Theorem and the Jarzynski Equation apply when far-from-equilibrium events take place *in an equilibrium bath with a temperature*. In many experiments and real-life systems, the bath is the very *source of nonequilibrium*. The Fluctuation Theorem and the Jarzynski Equation are of no help in that case and new theory needs to be developed. An obstacle here is constituted by the fact that there is no equivalent of temperature for the Lévy-stable distribution of velocities that is commonly associated with the nonequilibrium bath. For a Gaussian velocity distribution, the standard deviation is proportional to the square root of the temperature. However for a Lévy-stable distribution, the standard deviation diverges and, technically, the temperature works out to be infinite. In this article we have tried to contribute to the development of description and understanding of what can happen in nonequilibrium baths.

As was explained in the Introduction, baths consisting of Lévy particles lead to similar physics as baths in which active particles are suspended. In both cases there is a continuous input of energy into the system and there is no longer a Fluctuation-Dissipation Theorem to guide the understanding and description. Swimming bacteria are a prime example of active particles. That swimming *Escherichia coli* bacteria can indeed be accumulated in cavities as has been experimentally demonstrated [40].

In a recent paper, results are presented of a numerical simulation of an active-particlescontaining liquid [41]. A passive particle in this liquid was followed as a probe. This passive particle turned out to display Lévy-stable distributed displacements. What was simulated in this work was merely the Navier–Stokes equations and that passive particles exhibit these Lévy-stable distributed displacements is therefore a purely hydrodynamic effect due to active-particle-activity. That interesting and unexpected hydrodynamics can develop in liquids with immersed swimming bacteria has also been experimentally established [42].

The density profiles in Figures 2 and 5 are mindful of the coffee ring effect. When a coffee drop on a surface evaporates, the stain that is left behind is darkest towards the edge [43]. This effect is common with liquids that carry solutes. There are technological applications where it is important to control the coffee ring effect. The simple explanation for the effect goes as follows. The drop has the shape of a disk. It has a fixed radius and the height of the drop vanishes near the edge. With a uniform evaporation across the surface area of the drop, there must be a net outward radial flow to replenish lost fluid near the contact line. Solute is carried along with this flow and ultimately deposited near the contact line. Much theoretical, numerical, and experimental research has been devoted to the effect in the last quarter century (see Ref. [44] and references therein). It is common to use equilibrium concepts like Einstein's Fluctuation-Dissipation theorem when trying to account for the phenomenon. However, an evaporating drop is not in a thermodynamic equilibrium. It is certainly possible that solute particles exhibit the large jumps that are commonly encountered in nonequilibrium systems. The accumulation at the edge could then also be due to the mechanism that we discussed in this article.

In plasma physics, it is common to assume that the particles in a dense plasma follow the well-known Maxwell–Boltzmann distribution for particle speeds [1]. However, this equilibrium assumption may not always be valid, especially if the plasma is short lived and associated with an energy pulse. At Lawrence Livermore Lab, a table-top-size construction was developed to generate pulses of fast neutrons from high-energy deuterium collisions in plasma. Such collisions lead to the nuclear reaction D + D →3He + n [45]. In the experiments, it appears that the number of produced neutrons exceeds the theoretical predictions by more than an order of magnitude. The reason for this is most likely that the Maxwell–Boltzmann distribution only applies at thermodynamic equilibrium.

Plasmas in which energy is converted or transferred are of course not in a thermodynamic equilibrium. In containers with plasma, a homogeneous distribution is therefore unlikely and accumulation at the edge as described in this work is possible. This is important because it means that fusion reactions in a plasma will occur at different rates at different positions. Through feedback mechanisms, such inhomogeneities may rapidly augment and possibly develop into serious instabilities.

Engineered microswimmers is probably the field where our results could ultimately be most applicable. There are good methods and technologies for manipulating suspended micrometer size particles from the outside with acoustic, magnetic, or optic signals (see e.g., Refs. [46,47]). Today the exciting new developments are in the medical application of such microrobots. Clinical uses for imaging, sensing, targeted drug delivery, microsurgery, and artificial insemination are envisaged and researched [48]. The microswimmers and microrobots are particles that are operating in a very noisy environment. Accumulations as described and explained in this article are likely to be encountered.

**Author Contributions:** Conceptualization, S.Y. and M.B.; methodology, S.Y. and M.B.; software, S.Y.; formal analysis, S.Y. and M.B.; investigation, S.Y. and M.B.; writing—original draft preparation, M.B.; writing—review and editing, M.B.; supervision, M.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** We are grateful to Roland Winkler and Łukasz Ku´smierz for useful feedback as we were working on this article. We are also grateful to John Michael Jones and Keith Thomson for their help with the simulations that led to Figure 4.

**Conflicts of Interest:** The authors declare no conflict of interest.
