Thermodynamics of Currents in Nonequilibrium Diffusive Systems: Theory and Simulation

Abstract

Understanding the physics of nonequilibrium systems remains as one of the major challenges of modern theoretical physics. We believe nowadays that this problem can be cracked in part by investigating the macroscopic fluctuations of the currents characterizing nonequilibrium behavior, their statistics, associated structures and microscopic origin. This fundamental line of research has been severely hampered by the overwhelming complexity of this problem. However, during the last years two new powerful and general methods have appeared to investigate fluctuating behavior that are changing radically our understanding of nonequilibrium physics: a powerful macroscopic fluctuation theory (MFT) and a set of advanced computational techniques to measure rare events. In this work we study the statistics of current fluctuations in nonequilibrium diffusive systems, using macroscopic fluctuation theory as theoretical framework, and advanced Monte Carlo simulations of several stochastic lattice gases as a laboratory to test the emerging picture. Our quest will bring us from (1) the confirmation of an additivity conjecture in one and two dimensions, which considerably simplifies the MFT complex variational problem to compute the thermodynamics of currents, to (2) the discovery of novel isometric fluctuation relations, which opens an unexplored route toward a deeper understanding of nonequilibrium physics by bringing symmetry principles to the realm of fluctuations, and to (3) the observation of coherent structures in fluctuations, which appear via dynamic phase transitions involving a spontaneous symmetry breaking event at the fluctuating level. The clear-cut observation, measurement and characterization of these unexpected phenomena, well described by MFT, strongly support this theoretical scheme as the natural theory to understand the thermodynamics of currents in nonequilibrium diffusive media, opening new avenues of research in nonequilibrium physics.

Appendices

Appendix A: Additivity Predictions for Current Fluctuations in the 2D KMP Model

In this appendix we solve the simplified variational problem posed by MFT after the additivity conjecture of Sect. 5 for the particular case of the two-dimensional Kipnis-Marchioro-Presutti model defined in Sect. 3, when subject to a temperature gradient along the x-direction (i.e. in contact with thermal baths at the left and right boundaries at temperatures T _L and T _R , respectively) and with periodic boundary conditions along the y-direction. Due to the symmetry of the problem, the calculations here described are also useful for the 1D KMP model subject to the same temperature gradient, for which predictions follow by simply making zero all y-components.

The equation for the optimal density field under the additivity conjecture reads

$$ D[\rho_{\mathbf{J}}]^2 (\boldsymbol{\nabla}\rho_{\mathbf{J}})^2= \mathbf{J}^2 \bigl(1+2\sigma[\rho _{\mathbf{J}}]K\bigl(\mathbf{J}^2 \bigr) \bigr) $$

(111)

where K(J ²) is a constant of integration which guarantees the correct boundary conditions for ρ _J (r), which in this particular case are

$$ \rho_{\mathbf{J}}(x=0,y)=T_L ; \quad\quad\rho_{\mathbf{J}}(x=1,y)=T_R ; \quad\quad\rho_{\mathbf{J}}(x,y=0)=\rho_{\mathbf{J}}(x,y=1) . $$

(112)

Moreover, for the KMP model recall that the diffusivity and mobility transport coefficients entering MFT are D[ρ]=1/2 and σ[ρ]=ρ ², respectively. Interestingly, the optimal field solution of Eq. (111) depends exclusively on the modulus of the current vector J=|J|, that is ρ _J (r)=ρ _J (r). In addition, the symmetry of the problem suggests that ρ _J (r) depends exclusively on x, with no structure in the y-direction, compatible with the presence of an external gradient along the x-direction, i.e. ρ _J (r)≡ρ _J (x). In the last expression we have simplified our notation in order not to clutter formulas below. Under these considerations, the differential equation (111) for the optimal profile in the 2D KMP model becomes

$$ \biggl( \frac{d\rho_{J}(x)}{dx} \biggr)^2=4J^2 \bigl(1+2K(J) \rho _{J}^2(x) \bigr) $$

(113)

Here two different scenarios appear. On one hand, for large enough K(J) the r.h.s. of Eq. (113) does not vanish ∀x∈[0,1] and the resulting profile is monotone. In this case, and assuming T _L >T _R henceforth without loss of generality,

$$ \frac{d \rho_{J}(x)}{dx} = - 2J \sqrt{1+2\rho_{J}^2(x) K(J)} . $$

(114)

On the other hand, for K(J)<0 the r.h.s. of Eq. (113) may vanish at some points, resulting in a ρ _J (x) that is non-monotone and takes an unique value $\rho_{J}^{*}\equiv\sqrt{-1/2K(J)}$ in the extrema. Notice that the r.h.s. of Eq. (113) may be written in this case as $4J^{2}[1-(\rho_{J}(x)/\rho_{J}^{*})^{2}]$. It is then clear that, if non-monotone, the profile ρ _J (x) can only have a single maximum because: (i) $\rho_{J}(x)\le\rho_{J}^{*}$ ∀x∈[0,1] for the profile to be a real function, and (ii) several maxima are not possible because they should be separated by a minimum, which is not allowed because of (i). Hence for the non-monotone case (recall T _L >T _R )

$$\begin{aligned} \frac{d \rho_{J}(x)}{dx} = \left \{ \begin{array}{l@{\quad}l} +2J \sqrt{1- (\frac{\rho_{J}(x)}{\rho _J^*} )^2} ,& { x<x^* } \\ - 2J \sqrt{1- (\frac{\rho_{J}(x)}{\rho _J^*} )^2} ,& { x>x^* } \end{array} \right . \end{aligned}$$

(115)

where x ^∗ locates the profile maximum. This leaves us with two separated regimes for current fluctuations, with the crossover happening for $J_{0}\equiv\frac{T_{L}}{2} [\frac{\pi}{2} - \sin ^{-1} (\frac{T_{R}}{T_{L}} ) ]$. This crossover current can be obtained from Eq. (121) below by letting $\rho_{J}^{*}\rightarrow T_{L}$

1.1 A.1 Region I: Monotonous Regime (J<J ₀)

In this case, using Eq. (114) to change variables in Eq. (38) we have

$$ G(\mathbf{{J}})=\int_{T_L}^{T_R} d\rho_{J} \frac{1}{4J\rho_{J}^2\sqrt {1+2K(J)\rho_{J}^2}} \Bigl[ \Bigl(J_x-J\sqrt{1+2K(J) \rho_{J}^2} \Bigr)^2+J_y^2 \Bigr] , $$

(116)

with J _α the α component of vector J. This results in

$$ G(\mathbf{{J}})=\frac{J_x}{2} \biggl(\frac{1}{T_R}-\frac{1}{T_L} \biggr)-J^2K(J)+\frac{J}{2} \biggl[\frac{\sqrt{1+2K(J)T_L^2}}{T_L}- \frac {\sqrt{1+2K(J)T_R^2}}{T_R} \biggr] $$

(117)

Notice that, for ρ _J (x) to be monotone, $1+2K(J)\rho_{J}^{2}>0$. Thus, $K(J)>-(2T_{L}^{2})^{-1}$. Integrating now Eq. (114) we obtain the following implicit equation for ρ _J (x) in this regime

$$\begin{aligned} 2xJ = \left \{ \begin{array}{l@{\quad}l} {\frac{1}{\sqrt{2K(J)}} \ln [ \frac{ T_L + \sqrt {T_L^2 + \frac{1}{2K(J)}}}{ \rho_{J}(x) + \sqrt{\rho_{J}(x)^2 + \frac{1}{2K(J)}}} ] }, & { K(J)>0 } \\ \frac{\sin^{-1} [T_L\sqrt{-2K(J)} ] - \sin ^{-1} [\rho_{J}(x)\sqrt{-2K(J)} ]}{\sqrt{-2K(J)}} , & {-\frac{1}{2T_L^2}<K(J)<0 } \end{array} \right . \end{aligned}$$

(118)

Making x=1 and ρ _J (x=1)=T _R in the previous equation, we obtain the implicit expression for the constant K(J). To get a feeling on how it depends on J, note that in the limit $K(J)\rightarrow(-1/2T_{L}^{2})$, the current J→J ₀, while for K(J)→∞ one gets J→0. In addition, from Eq. (118) we see that for K(J)→0 we find J=(T _L −T _R )/2=〈J〉.

Sometimes it is interesting to work with the Legendre transform of the current LDF [1–7, 10, 24, 26], μ(λ)=max _J [G(J)+λ⋅J], where λ is a vector parameter conjugate to the current. Using the previous results for G(J) is easy to show [26] that μ(λ)=−K(λ)J ^∗(λ)², where J ^∗(λ) is the current associated with a given λ, and the constant K(λ)=K(|J ^∗(λ)|). The expression for J ^∗(λ) can be obtained from Eq. (118) above in the limit x→1. Finally, the optimal profile for a given λ is just $\rho_{\boldsymbol{\lambda}}(x)=\rho_{|\mathbf{J}^{*}(\boldsymbol{\lambda})|}(x)$.

1.2 A.2 Region II: Non-monotonous Regime (J>J ₀)

In this case the optimal profile has a single maximum $\rho_{J}^{*}\equiv \rho_{J}(x=x^{*})$ with $\rho_{J}^{*}=1/\sqrt{-2K(J)}$ and $-1/2T_{L}^{2}<K(J)<0$. Splitting the integral in Eq. (38) at x ^∗, and using now Eq. (115) to change variables, we arrive at

$$\begin{aligned} G(\mathbf{{J}}) = & \frac{J_x}{2} \biggl(\frac{1}{T_R}- \frac{1}{T_L} \biggr) - \frac{J}{2} \biggl[\frac{1}{T_L}\sqrt {1- \biggl(\frac{T_L}{\rho _J^*} \biggr)^2} + \frac{1}{T_R} \sqrt{1- \biggl(\frac{T_R}{\rho_J^*} \biggr)^2} \\ &{}- \frac{1}{2\rho_J^*} \biggl(\pi-\sin^{-1} \biggl( \frac{T_L}{\rho _J^*} \biggr)-\sin^{-1} \biggl(\frac{T_R}{\rho_J^*} \biggr) \biggr) \biggr] . \end{aligned}$$

(119)

Integrating Eq. (115) one gets an implicit equation for the non-monotone optimal profile

$$\begin{aligned} 2xJ = \left \{ \begin{array}{l@{\quad}l} {\rho_J^* [ \sin^{-1} ( \frac{\rho_{J}(x)}{\rho _J^*} ) - \sin^{-1} ( \frac{T_L}{\rho_J^*} ) ] } & \text{for } 0\le x<x^* \\ {2J + \rho_J^* [ \sin^{-1} ( \frac{T_R}{\rho _J^*} ) - \sin^{-1} ( \frac{\rho_{J}(x)}{\rho_J^*} ) ] } & \text{for } x^*<x\le1 \end{array} \right . \end{aligned}$$

(120)

At x=x ^∗ both branches of the above equation must coincide, and this condition provides simple equations for both x ^∗ and $\rho_{J}^{*}$

$$ J=\frac{\rho_J^*}{2} \biggl[\pi-\sin^{-1} \biggl(\frac{T_L}{\rho _J^*} \biggr)-\sin^{-1} \biggl(\frac{T_R}{\rho_J^*} \biggr) \biggr] ;\quad\quad x^*= \frac{\frac{\pi}{2}-\sin^{-1} (\frac{T_L}{\rho_J^*} ) }{\pi-\sin^{-1} (\frac{T_L}{\rho_J^*} )-\sin^{-1} (\frac{T_R}{\rho_J^*} )} . $$

(121)

Finally, as in the monotone regime, we can compute the Legendre transform of the current LDF, obtaining as before μ(λ)=−K(λ)J ^∗(λ)², where J ^∗(λ) and the constant K(λ)=K(|J ^∗(λ)|) can be easily computed from the previous expressions [24]. Note that, in λ-space, monotone profiles are expected for $|\boldsymbol{\lambda}+\boldsymbol{\epsilon}| \leq\frac{1}{2 T_{R}}\sqrt{1- (\frac{T_{R}}{T_{L}} )^{2}}$, where $\boldsymbol{\epsilon}=(\frac{1}{2} (T_{L}^{-1}-T_{R}^{-1} ),0)$ is the driving force in this case, while non-monotone profiles appear for $\frac{1}{2 T_{R}}\sqrt{1- (\frac{T_{R}}{T_{L}} )^{2}} \leq|\boldsymbol{\lambda}+\boldsymbol{\epsilon}| \leq\frac{1}{2} (\frac{1}{T_{L}}+\frac{1}{T_{R}} )$.

Figure 10 shows the predicted G(J) (left panel) and its Legendre transform (right panel) for the 2D-KMP model. Notice that the LDF is zero for J=〈J〉=((T _L −T _R )/2,0) and negative elsewhere. For small current fluctuations, J≈〈J〉, G(J) obeys the following quadratic form

$$ G(\mathbf{J})\approx-\frac{1}{2} \biggl(\frac{(J_x-(T_L-T_R)/2)^2}{\sigma _x^2}+\frac{J_y^2}{\sigma_y^2} \biggr) , $$

(122)

with $\sigma_{x}^{2}=(T_{L}^{2}+T_{L}T_{R}+T_{R}^{2})/3$ and $\sigma_{y}^{2}=T_{L} T_{R}$, resulting in Gaussian statistics for currents near the average as expected from the central limit theorem. A similar expansion for the Legendre transform yields

$$ \mu(\boldsymbol{\lambda})\approx\frac{\lambda_x}{2} \bigl[(T_L-T_R)+ \sigma_x^2\lambda _x \bigr]+ \frac{\sigma_y^2}{2}\lambda_y^2. $$

(123)

Notice that beyond this restricted Gaussian regime, current statistics is in general non-Gaussian. In particular, for large enough current deviations, G(J) decays linearly, meaning that the probability of such fluctuations is exponentially small in J (rather than J ²).

Fig. 10

Left panel: G(J) for the 2D-KMP model for T _L =2 and T _R =1. The blue circle signals the crossover from monotone (J<J ₀≡π/3) to non-monotone (J>π/3) optimal profiles. The green surface corresponds to the Gaussian approximation for small current fluctuations. Right panel: Legendre transform of both G(J) (brown) and G(J→〈J〉) (green), together with the projection in λ-space of the crossover between monotonous and non-monotonous regime (Color figure online)

Despite the complex structure of G(J) in both regime I and II above, it can be easily checked that for any pair of isometric current vectors J and J′, such that |J|=|J′|, the current LDF obeys

$$ G(\mathbf{J})-G\bigl(\mathbf{J}'\bigr)=\boldsymbol{\epsilon}\cdot\bigl(\mathbf{J}- \mathbf{J}'\bigr) . $$

(124)

This relation, known as Isometric Fluctuation Relation, is not particular for the 2D KMP model but a general results for time-reversible diffusive systems, see Sect. 6.

Appendix B: Constants of Motion

A sufficient condition for the Isometric Fluctuation Relation (IFR), Eqs. (58) or (124), to hold is that

$$ \frac{\delta\boldsymbol{\pi}_1[\rho(\mathbf{r})]}{\delta\rho(\mathbf{r}')} = 0 , $$

(125)

with the functional π ₁[ρ(r)] defined in Eq. (40) above. We have shown that condition (125) follows from the time-reversibility of the dynamics, in the sense that the evolution operator in the Fokker-Planck formulation of Eq. (1) obeys a local detailed balance condition, see Eq. (55). Condition (125) implies that π ₁[ρ(r)] is in fact a constant of motion, ϵ, independent of the profile ρ(r). Therefore we can use an arbitrary profile ρ(r), compatible with boundary conditions, to compute ϵ. We now choose boundary conditions to be gradient-like in the $\hat {x}$-direction, with densities ρ _L and ρ _R at the left and right reservoirs, respectively, and periodic boundary conditions in all other directions. Given these boundaries, we now select a linear profile

$$ \rho(\mathbf{r})=\rho_L + (\rho_R-\rho_L) x , $$

(126)

to compute ϵ, with x∈[0,1], and assume very general forms for the current and mobility functionals

$$\begin{aligned} \mathbf{Q} \bigl[\rho(\mathbf{r}) \bigr] \equiv& D_{0,0}[\rho] \boldsymbol{\nabla}\rho+ \sum _{n,m>0} D_{nm}[\rho] \bigl( \boldsymbol{\nabla}^m \rho \bigr)^{2n} \boldsymbol{\nabla}\rho , \\ \sigma \bigl[\rho(\mathbf{r}) \bigr] \equiv& \sigma_{0,0}[\rho] + \sum _{n,m>0} \sigma _{nm}[\rho] \bigl( \boldsymbol{\nabla}^m \rho \bigr)^{2n} , \end{aligned}$$

where as a convention we denote as F[ρ] a generic functional of the profile but not of its derivatives. It is now easy to show that $\boldsymbol{\epsilon}=\varepsilon\hat{x}+\mathbf{E}$, with

$$ \varepsilon= \int_{\rho_L}^{\rho_R} d \rho \frac{D_{0,0}(\rho)+\sum_{n>0} D_{n1}(\rho)(\rho_R-\rho_L)^{2n}}{ \sigma_{0,0}(\rho)+\sum_{m>0} \sigma_{m1}(\rho)(\rho_R-\rho_L)^{2m}} , $$

(127)

and $\hat{x}$ the unit vector along the gradient direction. In a similar way, if the following condition holds

$$ \frac{\delta}{\delta\rho(\mathbf{r}')} \int\mathbf{Q}\bigl[\rho(\mathbf{r})\bigr] d\mathbf{r}= 0 , $$

(128)

together with time-reversibility, Eq. (125), the system can be shown to obey an extended Isometric Fluctuation Relation which links any current fluctuation J in the presence of an external field E with any other isometric current fluctuation J′ in the presence of an arbitrarily-rotated external field E ^∗, and reduces to the standard IFR for E=E ^∗, see Eq. (79) in the main text. Condition (128) implies that ν≡∫Q[ρ(r)]d r is another constant of motion, which can be now written as $\boldsymbol{\nu}=\nu\hat{x}$, with

$$ \nu= \int_{\rho_L}^{\rho_R} d \rho \biggl[D_{0,0}( \rho)+\sum_{n>0} D_{n1}(\rho) ( \rho_R-\rho_L)^{2n} \biggr] , $$

(129)

As an example, for a diffusive system Q[ρ(r)]=−D[ρ]∇ ρ(r), with D[ρ] the diffusivity functional, and the above equations yield the familiar results

$$\begin{aligned} \varepsilon = & \int_{\rho_R}^{\rho_L} \frac{D(\rho)}{\sigma(\rho )}d\rho , \\ \nu = & \int_{\rho_R}^{\rho_L} D(\rho)d\rho , \end{aligned}$$

for a standard local mobility σ[ρ].