# Generalized Stochastic Fokker-Planck Equations

## Abstract

**:**

## 1. Introduction

## 2. Overdamped Brownian Particles with Long-Range Interactions

#### 2.1. The N-body Smoluchowski Equation

**r**

_{1},…,

**r**

_{N}) is the potential of interaction, and

**R**

_{i}(t) is a Gaussian white noise satisfying ⟨

**R**

_{i}(t)⟩ = 0 and $\langle {R}_{i}^{\alpha}\left(t\right){R}_{j}^{\beta}\left({t}^{\prime}\right)\rangle ={\delta}_{ij}{\delta}_{\alpha \beta}\delta \left(t-{t}^{\prime}\right)$, where i = 1, …, N labels the particles and α = 1,…, d labels the coordinates of space. The coefficients of diffusion and mobility are related to each other by the Einstein relation

_{N}(

**r**

_{1},…,

**r**

_{N}, t) is governed by the N-body FP equation [14]:

_{N}d

**r**

_{1}…d

**r**

_{N}= 1. In order to obtain Equation (4), we have used the Einstein relation (2) and we have defined β = 1/(k

_{B}T). The N-body distribution (4) corresponds to the statistical equilibrium state of the Brownian particles in the canonical ensemble. It gives the probability density of the microstate {

**r**

_{1},…,

**r**

_{N}}.

_{N}] satisfying the normalization condition constraint. Indeed, the cancelation of the first variations δF − µδI = 0, where µ is a Lagrange multiplier, returns Equation (4), and the second variations ${\delta}^{2}F=\frac{1}{2}{\displaystyle \int \frac{{(\delta {P}_{N})}^{2}}{{P}_{N}}}\phantom{\rule{0.2em}{0ex}}d{\mathbf{r}}_{1}\dots d{\mathbf{r}}_{N}>0$ are positive. At equilibrium, substituting Equation (4) in Equation (9), we get F[P

_{N}] = F(T) where F(T) = −k

_{B}T ln Z(T).

**Remark 1.**We have derived the canonical distribution (4) from the N-body stochastic dynamics (1). For systems with short-range interactions, one usually proceeds the other way round. We first derive the canonical distribution from the microcanonical distribution by considering a subsystem of a large system. Then, we introduce a Langevin dynamics that reproduces the canonical distribution at equilibrium. However, for systems with long-range interactions, we cannot proceed in this manner because the canonical distribution cannot be derived from the microcanonical distribution since the energy is non-additive [48]. Still, the canonical distribution is perfectly well-defined for systems with long-range interactions described at the start by a Langevin dynamics instead of a Hamiltonian dynamics.

#### 2.2. Long-Range Interactions

_{ij}= u(|

**r**

_{i}−

**r**

_{j}|). We assume furthermore that the potential of interaction is long-ranged which means that it decays at large distances as r

^{−γ}with γ < d [48]. For systems with long-range interactions, it can be shown that the correlations between the particles can be neglected in a proper thermodynamic limit N → +∞ [48]. Therefore, the mean field approximation becomes exact in this limit and the N-body distribution function can be factorized in a product of N one-body distribution functions

**r**, t) = NmP

_{1}(

**r**, t) is the mean density,

**Remark 2.**In Equation (13) the mean field potential appears as a convolution product (Φ = u * ρ) and the factor 1/2 in the energy of interaction (14) is introduced in order to avoid double counting. The particles may also experience an external potential Φ

_{ext}(

**r**). The energy of the particles in the external potential is W

_{ext}= ∫ ρΦ

_{ext}dr.

#### 2.3. Complex Systems: Generalized Thermodynamics

**r**) is the Boltzmann entropy. In this paper, we consider the case of “complex” systems for which small-scale constraints act on the particles (in addition to the long-range interaction) and modify their simple dynamics. We shall not try to model these constraints because they would lead to very complicated equations of motion that, most of the time, cannot be written explicitly. Therefore, these small-scale constraints appear as “hidden constraints”. We shall take them into account indirectly by using a form of “generalized entropy”. The fact that complex systems exhibit non-Boltzmannian distributions and non-Boltzmannian entropies has been observed in a wide variety of situations [19].

#### 2.4. The Free Energy F [ρ]

**r**

_{1},…,

**r**

_{N}} (any) with energy U(

**r**

_{1},…,

**r**

_{N}) is proportional to ${e}^{-\beta U}{}^{\left({\mathbf{r}}_{1},\dots ,{\mathbf{r}}_{N}\right)}$. For complex systems, we assume that some microstates are forbidden in physical space because of microscopic constraints that we shall not try to describe (in a future contribution [49], we shall consider the case where some microstates are forbidden in phase space). We denote by Ω the set of physically accessible microstates in physical space. We assume that the probability of an accessible microstate {

**r**

_{1},…,

**r**

_{N}} ∈ Ω is given by Equation (4). On the other hand, P

_{N}(

**r**

_{1},…,

**r**

_{N}) = 0 for a forbidden microstate {

**r**

_{1},…,

**r**

_{N}} ∉ Ω. Therefore, we replace the partition function (5) by

**r**) in position space and denote by Ω[ρ] the number of microstates {

**r**

_{i}} corresponding to the macrostate ρ(

**r**). The smooth density ρ(

**r**) corresponds to the coarse-grained density of Section 4.1 denoted $\overline{\rho}(\mathbf{r})$ but, to simplify the notations, we do not write the bar on ρ. The unconditional entropy of the macrostate ρ(

**r**) is defined by S

_{0}[ρ] = k

_{B}ln Ω[ρ]. In principle, the number of complexions Ω[ρ] can be obtained by a combinatorial analysis. If there is no microscopic constraint (simple systems) then, for N ≫ 1, we obtain the Boltzmann entropy S

_{0}[ρ] = −k

_{B}∫ (ρ/m) ln (ρ/Nm) d

**r**. However, if some microstates are forbidden (complex systems), we will have instead S

_{0}[ρ] = −k

_{B}∫ C(ρ) d

**r**where S

_{0}[ρ] is a generalized entropy that can be different from the Boltzmann entropy. The function C(ρ) can take various forms. For future purposes, we assume that C(ρ) is convex (C″ > 0).

**r**

_{1},…,

**r**

_{N}} in Equation (16), we can integrate on the macrostates {ρ(

**r**)}. If we consider a system with long-range interactions, so that a mean field approximation applies at the thermodynamic limit N → + ∞, we obtain for N ≫ 1:

#### 2.5. Variational Principle

_{∗}is the global minimum of free energy at fixed mass. This corresponds to the most probable macrostate in the canonical ensemble. We therefore have to solve the minimization problem

**Remark 3.**The variational principle (24) where ρ

_{∗}is the global minimum of free energy at fixed mass determines the strict statistical equilibrium state of the system. For systems with long-range interactions, it is very relevant to consider also local minima of free energy at fixed mass. They correspond to metastable states. For systems with long-range interactions, these metastable states have tremendously long lifetimes, scaling as e

^{N}, so they can be considered as fully stable states in practice [50].

#### 2.6. Equilibrium States

_{B}T. Substituting Equation (13) in Equation (27), we find that the density is determined by the integro-differential equation

^{−1}(−x). We note that, at equilibrium, the density is a function of the potential: ρ = ρ(Φ). Taking the derivative of Equation (27) with respect to ρ, we get

**r**= 0.

**Remark 4.**if the system is only subjected to an external potential Φ

_{ext}(

**r**), the equilibrium state is given by ρ(

**r**) = F [βΦ

_{ext}(

**r**) + α]. The second variations of free energy are always positive

**Remark 5.**Equation (29) determines the equilibrium state from the knowledge of the generalized entropy. Inversely, in certain cases, we know the equilibrium state and we want to determine the corresponding entropy. If we prescribe the equilibrium state in the form ρ = F (βΦ + α), we have C′(ρ) = −F

^{−1}(ρ) so the generalized entropy is given by Equation (19) with [46].

## 3. Generalized Mean Field Fokker-Planck Equations

#### 3.1. Simple Systems: The Mean Field Smoluchowski Equation

**r**, t) = NmP

_{1}(

**r**, t) can be obtained as follows [14]. One starts from the N-body Smoluchowski Equation (3) and writes down the equivalent of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy for the reduced probability distributions P

_{j}(

**r**

_{1},…,

**r**

_{j}, t). The equation for the one-body distribution function is

_{2}(

**r**,

**r**′, t) = N(N − 1)m

^{2}P

_{2}(

**r**,

**r**′, t) is the two-body distribution function. One then closes the hierarchy of equations in the limit N → +∞ by using the mean field approximation (11). In that limit, ρ

_{2}(

**r**,

**r**′, t) = ρ(

**r**, t)ρ(

**r**′, t). This leads to the mean field Smoluchowski equation

**r**, t) is given by Equation (13). Using the Einstein relation (2), the mean field Smoluchowski equation can be rewritten as

_{B}[ρ] at fixed mass. Therefore, the mean field Smoluchowski equation is fully consistent with Boltzmann’s thermodynamics.

#### 3.2. Complex Systems: The Generalized Mean Field Smoluchowski Equation

#### 3.3. Gradient Flow

**r**obtained by general arguments in Section 2.4 to the functions h(ρ) and g(ρ) occurring in the GFP equation (44) and, correspondingly, to the functions a(ρ) and b(ρ) determining the microscopic process underlying the dynamics (see [24] and Section 2.11 of [46]). Since h and g are positive in general, the function C is convex. We note that there exists an infinity of FP equations with the same entropy C(ρ) but different functions g(ρ) and h(ρ), namely all the equations with the same ratio h(ρ)/g(ρ) [46].

**J**, the mass M = ∫ ρ d

**r**is conserved provided that the current vanishes at infinity or that the normal component of the current vanishes on the boundary of the system.

#### 3.4. Generalized H-Theorem

**J**= 0, i.e., ∇(δF/δρ) = 0. Therefore, a steady state is determined by the condition (26), where µ is a constant of integration which plays the role of a chemical potential. This condition is equivalent to Equations (27)–(29). Therefore, a steady state of the GFP equation (50) is a critical point of free energy F at fixed mass. Using Lyapunov’s direct method [45], one can show that a steady state of the GFP equation (50) is dynamically stable if, and only if, it is a (local) minimum of F at fixed mass (maxima or saddle points of F are dynamically unstable). In this sense, dynamical and generalized thermodynamical stability in the canonical ensemble coincide. In general, the GFP equation relaxes towards a stable steady state for t → +∞ and stays there permanently. If several stable steady states exist, the choice of equilibrium is determined by a complicated notion of basin of attraction. Of course, there exist situations where the system has an even more complex dynamics. This is the case, for example, of self-gravitating Brownian particles (or bacterial populations) that can experience gravitational collapse (or chemotactic collapse) and never reach a steady state [16].

#### 3.5. Onsager’s Linear Thermodynamics

**Remark 6.**The GFP equation (50) can also be obtained from a variational principle called the principle of maximum dissipation of free energy (see Section 2.10.3 of [46]). Indeed, the current

**J**

_{∗}= −(1/ξ)g(ρ)∇(δF/δρ) minimizes the functional$\dot{F}+{E}_{d}$ where$\dot{F}[\mathbf{J}]={\displaystyle \int \mathbf{J}}\cdot \nabla (\delta F/\delta \rho )d\mathbf{r}$ and E

_{d}[

**J**] = ξ ∫

**J**

^{2}/(2g(ρ)) d

**r**. We have$\delta (\dot{F}+{E}_{d})=0$ and${\delta}^{2}\left(\dot{F}+{E}_{d}\right)=\xi {\displaystyle \int {\left(\delta \mathbf{J}\right)}^{2}/\left(2g\left(\rho \right)\right)}d\mathbf{r}>0$ (there is a misprint in [46])). Furthermore, ${E}_{d}\left[{\mathbf{J}}_{\ast}\right]=-\left(1/2\right)\dot{F}\left[{\mathbf{J}}_{\ast}\right]$.

#### 3.6. Particular Form: Normal Mobility and Generalized Diffusion

#### 3.7. Particular Form: Normal Diffusion and Generalized Mobility

#### 3.8. Generalized Smoluchowski Equation

#### 3.9. Expression in Terms of the Enthalpy

## 4. Theory of Fluctuations

#### 4.1. Simple Systems: The Stochastic Smoluchowski Equation

**r**, t) = ∑

_{i}δ(

**r**−

**r**

_{i}(t)) of particles satisfies a stochastic partial differential equation

_{d}(

**r**, t) is given by Equation (13) in which ρ(

**r**, t) is replaced by ρ

_{d}(

**r**, t). On the other hand,

**R**(

**r**, t) is a Gaussian white noise satisfying ⟨

**R**(

**r**, t)⟩ = 0 and ⟨R

^{α}(

**r**, t) R

^{β}(

**r**, t′)⟩ = δ

_{αβ}δ(t−t′)δ(

**r**−

**r**′), where α = 1,…,d labels the coordinates of space. This equation is exact and bears the same information as the N-body Langevin equations (1), or as the N-body Smoluchowski equation (3). In this sense, it contains too much information to be of practical use. Furthermore, ρ

_{d}(

**r**, t) is a sum of Dirac δ-functions, which is not easy to handle. If we take the ensemble, or noise, average of Equation (85), and make a mean field approximation, we recover the mean field Smoluchowski equation (37). However, in that case, we have lost the effect of fluctuations.

**r**, t) is replaced by $\overline{\rho}\left(\mathbf{r},t\right)$. This equation can be obtained from the theory of fluctuating hydrodynamics (see Appendix B of [15]). Although it has a mathematical form similar to Equation (85), this equation is fundamentally different from Equation (85) since it applies to a smooth density $\overline{\rho}\left(\mathbf{r},t\right)$, not to a sum of δ-functions. In a sense, it describes the evolution of the system at a mesoscopic level, intermediate between Equations (85) and (37).

#### 4.2. Complex Systems: The Generalized Stochastic Smoluchowski Equation

^{2}/2, i.e., $S=-\frac{1}{2}{\displaystyle \int {\rho}^{2}}d\mathbf{r}$. This is a Tsallis entropy of index q = 2 (see Section 5). We note that the function h(ρ) does not appear in the noise term.

**r**, t). The corresponding FP equation for the probability density P [ρ, t] of the density field ρ(

**r**, t) at time t is

#### 4.3. Particular Form: Normal Mobility and Generalized Diffusion

#### 4.4. Particular Form: Normal Diffusion and Generalized Mobility

#### 4.5. Equivalent Forms of the Generalized Stochastic Fokker-Planck Equation

## 5. A New Form of Generalized Entropy

^{γ−1}and g(ρ) = ρ(1 – Kρ/σ

_{0}). For γ ≠ 1 and σ

_{0}< +∞, we get a GFP equation with a power law diffusion and a density-dependent mobility taking into account exclusion (K > 0) or inclusion (K < 0) constraints in position space. It is associated with the generalized entropy S = −∫ C(ρ) d

**r**with [see equation (51)]:

^{γ}is that of a polytrope of index γ. The generalized Smoluchowski equation (71) with the same entropy as Equation (107) has an equation of state given by (see Equation (72)):

- For γ = 1 and σ
_{0}→ +∞, we recover the Smoluchowski equation which is a FP equation with a normal diffusion and a normal mobility [4]. It is associated with the Boltzmann entropy$$S=-{\displaystyle \int \rho \phantom{\rule{0.2em}{0ex}}\mathrm{ln}}\phantom{\rule{0.2em}{0ex}}\rho \phantom{\rule{0.2em}{0ex}}d\mathbf{r}.$$$$\rho ={e}^{-\beta \mathrm{\Phi}-\alpha -1}.$$ - For γ ≠ 1 and σ
_{0}→ +∞, we get a GFP equation with an anomalous diffusion and a constant mobility [30,31]. It is associated with the Tsallis entropy$$S=-\frac{1}{\gamma -1}{\displaystyle \int ({\rho}^{\gamma}-\rho )d\mathbf{r},}$$$$\rho -{\left[\frac{1}{\gamma}-\frac{\gamma -1}{\gamma}(\beta \mathrm{\Phi}+\alpha )\right]}_{+}^{1/(\gamma -1)}$$_{+}= x when x > 0 and [x]_{+}= 0 when x < 0. - For γ = 1 and σ
_{0}< +∞, we get the fermionic (K = +1) or bosonic (K = −1) Smoluchowski equation, which is a GFP equation with a normal diffusion and a variable mobility taking into account exclusion or inclusion constraints in position space [20,27–29,32,36,37,39,42,43,46,53]. It is associated with the Fermi-Dirac (K = +1) or Bose-Einstein (K = −1) entropy in position space$$S=-{\sigma}_{0}{\displaystyle \int \left\{\frac{\rho}{{\sigma}_{0}}\mathrm{ln}\frac{\rho}{{\sigma}_{0}}+\frac{1}{K}\left(1-\frac{K\rho}{{\sigma}_{0}}\right)\mathrm{ln}\left(1-\frac{K\rho}{{\sigma}_{0}}\right)\right\}d\mathbf{r}.}$$$$\rho -\frac{{\sigma}_{0}}{{e}^{\beta \mathrm{\Phi}+\alpha}+K}.$$$${P}_{GS}=-T\frac{{\sigma}_{0}}{K}\mathrm{ln}\left(1-K\rho /{\sigma}_{0}\right).$$ - For γ = 2, Equation (107) reduces to [46]:$$\xi \frac{\partial \rho}{\partial t}=\nabla \cdot [T\nabla {\rho}^{2}+\rho (1-K\rho /{\sigma}_{0})\nabla \mathrm{\Phi}].$$$$S=-\frac{2{\sigma}_{0}^{2}}{{K}^{2}}{\displaystyle \int \left(1-\frac{K\rho}{{\sigma}_{0}}\right)}\mathrm{ln}\left(1-\frac{K\rho}{{\sigma}_{0}}\right)d\mathbf{r}.$$$$\rho =\frac{{\sigma}_{0}}{K}{\left[1-{e}^{\frac{K}{2{\sigma}_{0}}\left(\beta \mathrm{\Phi}+\alpha \right)}\right]}_{+}.$$$${P}_{GS}=-2T\frac{{\sigma}_{0}}{K}\left[\rho +\frac{{\sigma}_{0}}{K}\mathrm{ln}(1-K\rho /{\sigma}_{0})\right].$$
- For γ → 0 and T → +∞ in such a way that γT is finite, and noted T again, Equation (107) becomes$$\xi \frac{\partial \rho}{\partial t}=\nabla \cdot [T\nabla \phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\rho +\rho (1-K\rho /{\sigma}_{0})\nabla \mathrm{\Phi}].$$
_{0}). It is associated with the generalized entropy$$S=-{\displaystyle \int \left[-\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\rho +\frac{K\rho}{{\sigma}_{0}}\mathrm{ln}\left(\frac{\rho /{\sigma}_{0}}{1-K\rho /{\sigma}_{0}}\right)+\mathrm{ln}(1-K\rho /{\sigma}_{0})\right]}\phantom{\rule{0.2em}{0ex}}d\mathbf{r}.$$$$\rho =\frac{{\sigma}_{0}/K}{1+W\left[\frac{1}{K}{e}^{\frac{{\sigma}_{0}}{K}\left(\beta \mathrm{\Phi}+\alpha \right)-1}\right]},$$^{W}= z. The generalized Smoluchowski equation (71) with the same entropy as Equation (124) has an equation of state$${P}_{GS}=T\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\left(\frac{\rho}{1-K\rho /{\sigma}_{0}}\right).$$_{0}→ +∞, the generalized entropy (125) reduces to the log-entropy [54]:$$S={\displaystyle \int \mathrm{ln}\phantom{\rule{0.2em}{0ex}}\rho \phantom{\rule{0.2em}{0ex}}d\mathbf{r},}$$$$\rho =\frac{1}{\beta \mathrm{\Phi}+\alpha}.$$

**r**, t) is given by Equation (13). Depending on the form of the potential of interaction, the steady states of Equation (107) exhibit a rich variety of phase transitions, as studied in [53] for self-gravitating systems and bacterial populations. When fluctuations are taken into account, the system exhibits random transitions from one phase to the other similarly to the study of [47].

## 6. Generalized Stochastic Cahn-Hilliard Equations

#### 6.1. Short-Range Interactions

**r**–

**r**′|) is a short-range potential of interaction but that the stochastic GFP equation (90) remains valid. This is the case for a potential u(|

**r**–

**r**′|) that is screened on a distance that is short with respect to the system size but large with respect to the characteristic microscopic scale. Setting

**q**=

**r**–

**r**′ and writing

**r**+

**q**, t) to second order in

**q**so that

^{1}

^{/}

^{2}has the dimension of a length corresponding to the range of the interaction. For the sake of generality, we assume that the particles are submitted, in addition to the self-interaction, to an external potential Φ

_{ext}(

**r**). In that case, the previously derived GFP equations remain valid provided that Φ is replaced by Φ + Φ

_{ext}. On the other hand, the energy is $E=\frac{1}{2}{\displaystyle \int \rho \mathrm{\Phi}\phantom{\rule{0.2em}{0ex}}d\mathbf{r}+{\displaystyle \int \rho {\mathrm{\Phi}}_{\mathrm{ext}}d\mathbf{r}}}$ so that the free energy (47) becomes

^{2}ρ = Ar

^{2}+ B, where we have defined k

^{2}= 2a/b, A = 2Φ

_{0}/b, and $B=-\left(2/b\right){\mathrm{\Phi}}_{0}{r}_{0}^{2}-2\mu /b$. The solution of this equation is

_{s}(

**r**) is the solution of the homogeneous Helmholtz equation Δρ

_{s}+k

^{2}ρ

_{s}=0. In the spherically symmetric case, ρ

_{s}(r) = K cos(kr) in d = 1, ρ

_{s}(r) = K J

_{0}(kr) in d = 2, and ρ

_{s}(r) = K sin(kr)/r in d = 3.

#### 6.2. Analogy with Cahn-Hilliard Equations

**r**, t) appears explicitly in the deterministic current and in the noise term of Equations (135) and (139) through the function g(ρ) while it is absent in the stochastic Cahn-Hilliard equation (270). Secondly, in the Cahn-Hilliard equation, the potential V(ρ) has a double-well shape of the typical form V(ρ) = A(σ

^{2}− ρ

^{2})

^{2}leading to a phase separation while, in the present case, the potential (137) is more general. It is only in the particular case g(ρ) = 1 and C(ρ) = ρ

^{4}, implying h(ρ) = 12ρ

^{2}, that Equation (135)–(136), or Equation (139), reduces to a form that is formally equivalent to the Cahn-Hilliard equation with $V(\rho )=\frac{2T}{b}{(\frac{a}{4T}-{\rho}^{2})}^{2}$. Finally, we note that the Cahn-Hilliard equation is a heuristic equation (to our knowledge it has not been derived from first principles or from a microscopic dynamics) while Equation (135)–(136), or Equation (139), is derived as a particular limit of the GFP equation (90). As a result, the potential V(ρ) in Equation (137) is entirely determined by the function C(ρ) entering in the generalized free energy defined by Equation (134).

#### 6.3. Expanded Form of the Generalized Stochastic Cahn-Hilliard Equation

#### 6.4. Particular Form: Anomalous Diffusion and Normal Mobility

#### 6.5. Particular Form: Normal Diffusion and Anomalous Mobility

#### 6.6. Equivalent Forms of the Generalized Stochastic Cahn-Hilliard Equation

#### 6.7. Simple Systems: Normal Diffusion and Normal Mobility

## 7. Analogy with An Effective Generalized Thermodynamics

#### 7.1. General Results

_{eff}(ρ), an external potential Φ

_{ext}(

**r**), and an effective temperature T

_{eff}. We note that the effective temperature T

_{eff}is positive when a < 0 (repulsive interactions) and negative when a > 0 (attractive interactions). The complete equation can be interpreted as a stochastic GFP equation. However, we note that the temperature appearing in the noise term is the thermodynamic temperature T, not the effective temperature T

_{eff}. The effective generalized entropy associated with Equation (171) is determined by the relation

**Remark 7.**When b ≠ 0, we can rewrite Equation (157) as

#### 7.2. Simple Systems

^{W}= z. In the case of attractive interactions (a > 0), W (z) is a new function defined implicitly by the equation W

^{−1}e

^{W}= z.

_{eff}(ρ), an external potential, and an effective temperature T

_{eff}. The effective generalized entropy associated with Equation (187) is

#### 7.3. Simple Systems at T = 0

_{eff}= −a and h

_{eff}(ρ) = ρ, we can rewrite Equation (191) as

_{eff}(ρ) = ρ, an external potential Φ

_{ext}(

**r**), and an effective temperature T

_{eff}. The effective generalized entropy associated with Equation (194) is

**Remark 8.**The situation is very similar to the one encountered in the case of the Gross-Pitaevskii (GP) equation (see [56] and Section II of [57]). Indeed, the GP equation can be derived from the mean field Schrödinger equation with a potential Φ(

**r**, t) = ∫ u(|

**r**−

**r**′|)ρ(r′, t) dr′ by using Equation (166) valid for systems with short-range interactions. Actually, Equation (166) corresponds to a pair contact potential of the form u = −aδ(

**r**–

**r**′). The coefficient a is related to the scattering length a

_{s}of the bosons by the relation a = −4πa

_{s}ħ/m

^{3}. We have a < 0 for repulsive self-interactions (a

_{s}> 0) and a > 0 for attractive self-interactions (a

_{s}< 0). Using the Madelung transformation, the GP equation can be written in the form of hydrodynamic equations involving a quantum potential and a pressure associated with a polytropic equation of state of index γ = 2 equivalent to Equation (197).

## 8. Application to Systems of Physical Interest

#### 8.1. Self-Gravitating Brownian Particles

_{d}is the surface of a d-dimensional hypersphere. If we take fluctuations into account, the evolution of the smooth density field is described by the stochastic Smoluchowski equation

^{−1}. In that case, the Poisson equation (200) is replaced by the screened Poisson equation

^{2}→ +∞), we can make the approximation Φ = −(S

_{d}G/k

^{2})ρ − (S

_{d}G/k

^{4})Δρ as in Section 6, and we get the generalized stochastic Cahn-Hilliard equation

**Remark 9.**The Smoluchowski-Poisson system (199)–(200) describes the evolution of Brownian particles that interact through the attractive gravitational force (G > 0). The case of Brownian charges that interact through the repulsive electric force leads to the Debye-Hückel [66], or Nernst-Planck [67–69], model. It corresponds to Equations (199)–(200) with G < 0. Similar Fokker-Planck equations have been introduced by Chavanis [70,71] for two-dimensional point vortices. The interaction between point vortices is “repulsive” at positive temperatures and “attractive” at negative temperatures [72]. We refer to [70,71] for more details about these systems and their analogies.

#### 8.2. Colloid Particles at a Fluid Interface

**r**, t) of colloids is governed by the system of equations

**r**, t) is the ensemble-averaged interfacial deformation, γ is the surface tension, and λ is the capillary length.

^{2}= 1/λ

^{2}, and 2πG = f

^{2}/γ. Using the formalism developed in the present paper, we can propose a generalized model of colloid particles at a fluid interface of the form

^{2}/γ)ρ + (fλ

^{4}/γ)Δρ as in Section 6, and we get the generalized stochastic Cahn-Hilliard equation

#### 8.3. Superconductor of Type-II

_{0}H

_{c}

_{2}/ρ

_{n}c

^{2}. Here, Φ

_{0}is the magnetic quantum flux, c is the speed of light, ρ

_{n}is the resistivity of the normal phase, and H

_{c}

_{2}is the upper critical field. The first term on the right hand side of Equation (211) represents the vortex-vortex interaction. It is given by $\mathbf{J}(\mathbf{r})=[{\mathrm{\Phi}}_{0}^{2}/(8\pi {\lambda}^{2})]{K}_{1}(|\mathbf{r}|/\lambda )\widehat{\mathbf{r}}$, where the function K

_{1}is a Bessel function decaying exponentially for |r| > λ, and λ is the London penetration length. The interaction is repulsive. The second term on the right hand side of Equation (211) accounts for the interaction between pinning centers, modeled as localized traps, and flux lines. Here G is the force due to a pinning center located at R

_{p}, l is the range of the wells (typically l ˝ λ), and p = 1, …, N

_{p}(N

_{p}is the total number of pinning centers). We shall regard this term as a (given) external force deriving from a potential U

_{ext}. Finally, the last term in Equation (211) is an uncorrelated thermal noise.

^{2}= 1/λ

^{2}, and $2\pi G=-{\mathrm{\Phi}}_{0}^{2}/(4{\lambda}^{2})$. Using the formalism developed in the present paper, we can propose a generalized model of superconductors of type II of the form

_{B}T and g(ρ) = ρ, and when the term $({\lambda}^{2}{\mathrm{\Phi}}_{0}^{2}/4)\mathrm{\Delta}\rho $ is neglected in the expansion of U (see Section 7), as well as the noise term, we recover the equation

#### 8.4. Dynamical Theory of Nucleation

**r**, t) is the local density field, D is the tracer diffusion constant for the large molecules in solution, F [ρ] is the free energy which is a functional of the local density (see Appendix C), β = 1/k

_{B}T where T is the temperature, and

**R**(

**r**, t) is a delta-correlated white noise. Equation (217) is equivalent to the generalized stochastic Smoluchowski equation (92) with a normal mobility. Using the theory developed in the present paper, we can propose a generalized model of nucleation of the form

#### 8.5. Chemotaxis of Bacterial Populations

**r**, t) and the evolution of the secreted chemical c(

**r**, t). The bacteria diffuse with a diffusion coefficient D and they also move along the gradient of the chemical (chemotactic drift). The coefficient χ is a measure of the strength of the influence of the chemical gradient on the flow of bacteria. The interaction can be attractive (χ > 0) or repulsive (χ < 0). On the other hand, the chemical is produced by the bacteria with a rate h and is degraded with a rate k. It also diffuses with a diffusion coefficient D

_{c}. Equations (219)–(220) can be derived from N-body stochastic Langevin equations in a mean field approximation (see [78] and Appendix A of [79]). If we take fluctuations into account, we obtain the stochastic KS model

_{c}is large, the term ∂c/∂t in Equation (220) can be neglected. In the case where there is no degradation of the chemical (k = 0), writing h = λD

_{c}and taking the limit D

_{c}→ +∞ with λ = O(1), we get the modified Poisson equation (see Appendix C of [79]):

_{c}and $k={k}_{0}^{2}{D}_{c}$, and taking the limit D

_{c}→ +∞ with λ = O(1) and k

_{0}= O(1), we get the screened Poisson equation (see Appendix C of [79]):

**r**, t) of the chemical plays the same role as the gravitational potential Φ(

**r**, t). We note that the field equation (220) in the KS model is more complicated than the Poisson equation (200) in the original SP system. When the term ∂c/∂t is taken into account in Equation (220), the proper expressions of the free energy and of the H-theorem are given in [46] and in Appendix E of [79]. On the other hand, in the framework of the KS model, it is possible to rigorously justify the modified Poisson equation (222) and the screened Poisson equation (223) that were introduced heuristically in the SP system (see Section 8.1). As a result, the KS model admits spatially homogeneous distributions while the original SP system does not. As emphasized in [63], this removes the so-called Jeans swindle [80] appearing in astrophysics.

**Remark 10.**As emphasized in [46], the chemotaxis of bacterial population is an important physical model described by generalized stochastic FP equations. It is associated with a notion of generalized thermodynamics because, in the case of complex systems (like those occurring in biology), the coefficients of diffusion and drift depend on the density. This is an heuristic approach to take into account microscopic constraints that affect the dynamics of particles at small scales and lead to non-Boltzmannian equilibrium distributions. Indeed, it is not surprising that the mobility or the diffusive properties of a bacterium depend on its environment. For example, in a dense medium, its motion can be hampered by the presence of the other bacteria so that its mobility is reduced.

#### 8.6. Application to 2D Turbulence

_{2})

^{−1/2}exp(−σ

^{2}/2Ω

_{2}), the generalized entropy is proportional to minus the enstrophy $S=-(1/2{\mathrm{\Omega}}_{2})\int {\overline{\omega}}^{2}d\mathbf{r}$ (see [87] and Section 5 of [88]). The functional $S[\overline{\omega}]$ has the status of an entropy in the sense of the theory of large deviations [84]. Indeed, the probability of the coarse-grained vorticity field $\overline{\omega}(\mathbf{r})$ at statistical equilibrium is given by

**r**and A is the domain area. By construction, these equations conserve the energy and the circulation and increase the generalized entropy until the equilibrium state is reached (H-theorem).

## 9. Conclusion

## A. Application of the Landau-Lifshitz Theory of Fluctuations

**J**is the current

**q**(

**r**, t). In order to use the general theory of fluctuations [51], we divide the fluid volume in small elements ΔV and take the average of each quantity in each element. The continuum limit ΔV → 0 will be performed in the final expressions. Equations (250) and (251) correspond to the equations

_{a}→ q

_{α}. The X

_{a}can be obtained from the expression of the rate of production of entropy. In fact, since we are working in the canonical ensemble, the proper thermodynamical potential is the free energy F = E − T S. Taking the time derivative of the free energy functional, using Equation (250), and integrating by parts, we obtain

_{a}are given by

_{ab}that appear in Equation (252). Comparing Equations (251), (252) and (257), we find that

**r**, t) satisfies

## B. Stochastic Ginzburg-Landau and Cahn-Hilliard Equations

**r**, t) is a one-dimensional Gaussian white noise. F[ρ] can be an arbitrary functional of ρ, but it is usually written in the form

**r**, t) at time t is

**R**(

**r**, t) is a d-dimensional Gaussian white noise. We note that M = ∫ ρ d

**r**is conserved. For a functional of the form of Equation (264), Equation (270) can be rewritten as

**r**, t) at time t is

## C. Long and Short-Range Interactions

_{LR}and a short-range potential u

_{SR}, so that u = u

_{LR}+ u

_{SR}. Concerning the long-range potential, we make the mean field approximation ρ

_{2}(

**r**,

**r**′, t) = ρ(

**r**, t)ρ(

**r**′, t) leading to

**r**, t) is the mean field potential defined by Equation (13) with u replaced by u

_{LR}. To evaluate the integral corresponding to the short-range interactions, we use an approximation that has become standard in the DDFT of fluids [92] and take

_{ex}[ρ] is the excess free energy calculated at equilibrium. Equation (277) is exact at equilibrium (see, e.g., [93]), and the approximation consists in extending it out-of-equilibrium with the actual density ρ(

**r**, t) calculated at each time. This closure is equivalent to assuming that the two-body dynamic correlations are the same as those in an equilibrium fluid with the same one-body density profile. With the approximations (276) and (277), Equation (35) becomes

_{ex}depends only on the density (and on the temperature T that is fixed in the canonical ensemble), we can write

_{id}(

**r**, t) = ρ(

**r**, t)k

_{B}T/m is the ideal gas law, P

_{ex}= P

_{ex}(ρ) is the excess pressure due to short-range interactions, and P = P

_{id}+ P

_{ex}is the total pressure given by a barotropic equation of state P = P (ρ). Substituting Equation (279) in Equation (278), we obtain the generalized Smoluchowski equation

_{ex}= 0), the pressure reduces to the perfect gas law P = ρk

_{B}T/m, and we recover the mean field Smoluchowski equation (37).

## Conflicts of Interest

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Chavanis, P.-H.
Generalized Stochastic Fokker-Planck Equations. *Entropy* **2015**, *17*, 3205-3252.
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Generalized Stochastic Fokker-Planck Equations. *Entropy*. 2015; 17(5):3205-3252.
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