On thermodynamics of stationary states of diffusive systems. Giovanni Jona-Lasinio Coauthors: L. Bertini, A. De Sole, D. Gabrielli, C.

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1 On thermodynamics of stationary states of diffusive systems Giovanni Jona-Lasinio Coauthors: L. Bertini, A. De Sole, D. Gabrielli, C. Landim Newton Institute, October 29, 2013

2 0. PRELIMINARIES 1. RARE FLUCTUATIONS 2. SOME PREDICTIONS 3. THERMODYNAMIC TRANSFORMATIONS 4. SUMMARY

3 0. PRELIMINARIES

4 Why studying irreversible processes is so much more difficult than understanding equilibrium phenomena? In equilibrium we have a well established phenomenological theory in terms of thermodynamic potentials which depend on few state variables. In non equilibrium we do not know how to define meaningful analogs of, e.g. the entropy, the free energy,...: even in the simplest situations any relevant macroscopic quantity depends both on the system and the environment.

5 In equilibrium statistical mechanics we do not have to solve any equation of motion and the Gibbs distribution provides the basis for the calculation of macroscopic quantities and their fluctuations. In non equilibrium we cannot bypass the dynamics even in the study of stationary states which we may consider as the simplest beyond equilibrium.

6 Over the last ten years, in collaboration with L.Bertini, A. De Sole, D. Gabrielli and C. Landim we have formulated a theory, now known as Macroscopic Fluctuation Theory (MFT), based on the study of rare fluctuations. This theory may be seen as an extension of Einstein equilibrium fluctuation theory to stationary nonequilibrium states combined with a dynamical point of view. The guide of lattice gas models has been crucial. MFT is very powerful in studying concrete microscopic models but can be used also as a phenomenological theory. The input it requires are measurable transport coefficients. It leads to several new interesting predictions.

7 Einstein theory of equilibrium fluctuations In Landau-Lifshitz book on statistical mechanics one finds the following formula for the probability of a fluctuation in a system in contact with an environment where P e R min T 0 (1) R min = E T 0 S + P 0 V (2) is the minimal work necessary to produce the fluctuation with a reversible transformation and E, S, V are the corresponding variations of energy, entropy and volume. T 0, P 0 are the temperature and pressure of the environment.

8 Typical setting

9 Assumptions 1. The macroscopic state is completely described by the local density ρ = ρ(t, x) and the associated current j = j(t, x). 2. The macroscopic evolution is given by the continuity equation together with the constitutive equation t ρ + j = 0 (3) j = J(t, ρ) = D(ρ) ρ + χ(ρ) E(t) (4) where the diffusion coefficient D(ρ) and the mobility χ(ρ) are d d positive matrices. The transport coefficients D and χ satisfy the local Einstein relation D(ρ) = χ(ρ) f 0 (ρ) (5) where f 0 is the equilibrium specific free energy.

10 The equations (3) (4) have to be supplemented by the appropriate boundary condition on Λ due to the interaction with the external reservoirs. If λ(t, x), x Λ is the chemical potential of the external reservoirs, this boundary condition is f 0( ρ(t, x) ) = λ(t, x) x Λ. (6)

11 Question: what is the probability of a macroscopic trajectory?

12 2. RARE FLUCTUATIONS

13 The large deviation formula for joint fluctuations of the density and current L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim, Phys. Rev. Lett. 94, (2005); J. Stat. Phys 123, 237 (2006) P ( ρ(t), j(t) t [0, T ] ) exp{ ɛ d G [0,T ] (ρ, j)} (7) Here P is the stationary probability measure, G [0,T ] (ρ, j) = { V (ρ(0)) + I[0,T ] (j) if t ρ + j = 0 + otherwise (8) where I [0,T ] (j) = 1 4 T 0 dt [j J(ρ)], χ(ρ) 1 [j J(ρ)] (9) in which we recall that J(ρ) = D(ρ) ρ + χ(ρ)e.

14 Time reversal To the time reversed process corresponds the adjoint generator with respect to the stationary ensemble. Let us define the operator inverting the time of a trajectory [θf](t) = f( t) for f scalar and [θj](t) = j( t) for the current. The stationary adjoint process, that we denote by P N,a, is the time reversal of P N, i.e. we have µ N µ N P N,a = P N ϑ 1. Then µ N µ N ) P N µ (ρ, j t [ T, T ] N ) = P (ϑρ, N,a ϑj t [ T, T ] (10) µ N At the level of large deviations this implies G [ T,T ] (ρ, j) = G[ T,T a ](ϑρ, ϑj) (11) where G[ T,T a ] process. is the large deviation functional for the adjoint

15 The quasi-potential and the associated Hamilton-Jacobi equation From the previous time reversal relation it follows that V (ρ) satisfies the following Hamilton-Jacobi equation δv δv δρ, χ(ρ) δv δρ δρ, J(ρ) = 0 (12) corresponding to the Hamiltonian H(ρ, π) = π χ(ρ) π where π is the conjugate momentum. + π J(ρ) (13)

16 The hydrodynamic equations in terms of V ( t ρ = χ(ρ) δv δρ ( t ρ = χ(ρ) δv δρ ) + A(ρ) (14) ) A(ρ) (15) The second equation is the hydrodynamics corresponding to the time reversed system. The Hamilton Jacobi equation implies the orthogonality condition δv δρ, A(ρ) = 0 Correspondingly there is an orthogonal decomposition of the current J(ρ) = J S (ρ) + J A (ρ) with J S (ρ) = χ(ρ) δv δρ, JS χ 1 J A = 0 (16)

17 As a simple illustration let us consider a charged particle in a viscous medium subjected to a magnetic field, ṗ = e mc p H 1 τ p, (17) where p is the momentum, e the charge, H the magnetic field, m the mass, c the velocity of the light, and τ the relaxation time. The dissipative term p/τ is orthogonal to the Lorenz force p H. We define time reversal as the transformation p p, H H. In this case the adjoint equation coincides with the time reversed dynamics, which is given by ṗ = e mc p H 1 τ p (18) In this example, J S (p) = p/τ and J A (p) = (e/mc)p H.

18 Another simple example is the case of a system of independent particles, the corrisponding transport coefficients are D(ρ) = I and χ(ρ) = ρi where D 0, χ 0 are scalar and I denotes the identity matrix. In the one dimensional case, with Λ = (0, L), λ(0) = λ 0, λ(l) = λ 1 the stationary density profile is ρ(x) = ρ 0 (1 x/l) + ρ 1 x/l where ρ 0 and ρ 1 are the densities associated to λ 0 and λ 1. In this case J S (ρ) = ρ + ρ 1 ρ 0 L J A (ρ) = ρ 1 ρ 0 ρ L ρ ρ ρ

19 2. SOME PREDICTIONS

20 1. - Nonequilibrium long range correlations 2. - Nonequilibrium phase transitions impossible in equilibrium 3. - Dynamical phase transitions in current flutuations with spontaneous symmetry breaking of translational invariance in time 4. - Universality of the moments in current fluctuations

21 Nonequilibrium phase transitions Bertini, De Sole, Gabrielli, Jona-Lasinio, Landim, J. Stat. Mech., L11001 (2010) Are there phase transitions forbidden in equilibrium but possible in nonequilibrium? We consider the weakly asymmetric simple exclusion process which is characterized at the macroscopic scale by D = 1 and χ = ρ(1 ρ). We fix boundary conditions on the segment [0, 1] ρ(0) = ρ 0 and ρ(1) = ρ 1. We take a constant external field E which may push the current in the same direction or opposite to the boundary conditions. The interesting case is the second one. For E sufficiently large there are density profiles where V (ρ) is non-differentiable. The hydrodynamic equation for this model is t ρ = ρ χ(ρ)e (19)

22 V (ρ) ρ Figure: Graph of the quasi-potential. ρ c is a caustic point. ρ c ρ

23 It can be shown that in the limit E =, if the density profile ρ is suitably chosen, the variational principle admits two minimizers. Then by a continuity argument one shows that this persists when the external field E is large. A ρ 0 0 ρ(x) y y 0 y + ρ 1 1 x Figure: Graph of a caustic density profile for E =. The shaded regions have equal area.

24 Thermodynamics of currents: the Φ(J) functional L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim, Phys. Rev. Lett. 94, (2005); J. Stat. Phys 123, 237 (2006) Let J(x) be the time average of j(t, x) that we assume divergence free, i.e. T J(x) = 1 j(x, t)dt (20) T 0 and define Φ(J) = lim inf 1 T j T I [0,T ](j), (21) where the infimum is carried over all paths j = j(t, u) having time average J. This functional is convex and satisfies a Gallavotti-Cohen type relationship Φ(J) Φ( J) = Φ(J) Φ a (J) = J, E + dσ λ 0 J ˆn (22) Note that the right hand side of (22) is the power produced by the external field and the boundary reservoirs. Entropy production can be simply derived from Φ(J)). Λ

25 Dynamical phase transitions There is a model of heat conduction introduced by Kipnis, Marchioro and Presutti characterized macroscopically by D = 1 and χ = ρ 2. One can show that in equilibrium with periodic boundary conditions and for small J Φ(J) = 1 J 2 2 ρ 2 (23) where ρ is the flat equilibrium profile. For J > J c, Φ(J) < 1 2 J 2 ρ 2 for a time dependent density profile which is a traveling wave of the form ρ 0 (x vt). Translation invariance in time is spontaneously broken.

26 Numerical simulation P. I. Hurtado, P.L. Garrido, Phys. Rev. Lett. 107, (2011)

27 Universality in current fluctuations C. Appert-Rolland, B. Derrida, V. Lecomte, F. van Wijland, Phys.Rev. E 78, (2008) Let Q(t) = t 0 j(t )dt the total integrated current during the time interval (0, t). Define the generating function of the cumulants of Q ln exp sq ψ J (s) = lim = Φ (s) (24) t t where the brackets denote an average over the time evolution during (0, t). Φ (s) is the Legendre transform of Φ(J). The authors estimate Φ(J) from the large deviation formula P({ρ(x, t), j(x, t)}) exp L 4 from which they obtain T 0 dt [j J(ρ)], χ(ρ) 1 [j J(ρ)] Q 2n 2n! lim = B 2n 2 t t n!(n 1)! D( χχ 8D 2 )n L 2n 2

28 3. THERMODYNAMIC TRANSFORMATIONS

29 Energy balance Fix T > 0, a density profile ρ(x), an external field E(t, x) and a chemical potential λ(t, x), 0 t T. Let ρ(t, x) the solution of hydrodynamics with initial condition ρ(x) and j(t, x) the corresponding current. The total energy involved in the process is T { } W [0,T ] = dt dσ(x) λ(t, x) j(t, x) ˆn(x)+ dx j(t, x) E(t, x), 0 Λ Λ (25) where ˆn is the outer normal to Λ and dσ is the surface measure on Λ. The first term on the right hand side is the energy provided by the reservoirs while the second is the energy provided by the external field.

30 Using the Einstein relation and the divergence theorem W [0,T ] can be written T W [0,T ] = F (ρ(t )) F (ρ(0)) + dt 0 Λ dx j(t) χ(u(t)) 1 j(t) (26) where F (ρ) = dx f(ρ(x)). From this equation the inequality follows Λ W [0,T ] F (ρ(t )) F (ρ(0)) (27) which is the second law here derived dynamically.

31 For very slow transformations taking into account the orthogonal decomposition of the current J(ρ) = J S (ρ) + J A (ρ) the second term in (26) can be written T 0 dt dx j S (t) χ(u(t)) 1 j S (t) + Λ T 0 dt dx j A (t) χ(u(t)) 1 j A (t) Λ (28) We identify the last term with the work necessary to keep the system out of equilibrium. This can be seen by recalling the hydrodynamic equation expressed in terms of V ( t ρ = χ(ρ) δv ) + A(ρ) (29) δρ Consider a stationary state. Since δv δρ coincides with J A. = 0 the stationary current

32 Thermodynamic interpretation of the quasi-potentia V (ρ) Consider the following transformation: at time t = 0 the system is in a stationary state ρ 0 (x) corresponding to a chemical potential λ 0 (x) which suddenly changes to λ 1 (x). The system will relax to a new stationary state ρ 1 (x) following hydrodynamics with new boundary conditions. W [0,T ] = F (ρ(t )) F ( ρ) + V λ1,e 1 ( ρ 0 ) V λ1,e 1 (ρ(t )) T + dt dx J A (ρ(t)) χ(ρ(t)) 1 J A (ρ(t)). 0 Λ (30) This equation shows that in the transformation considered the variation of the quasi potential in the interval [0, T ] represents the energy dissipated by the thermodynamic force δv λ,e(ρ) δρ.

33 Renormalized work L. Bertini, D. Gabrielli, G. Jona-Lasinio, C. Landim, (2012), J. Stat. Phys. 149, 773 (2012); Phys. Rev. Lett. 110, (2013). We define the renormalized work T W[0,T ren ] = F (ρ(t )) F (ρ(0)) + dt dx j S (t) χ(u(t)) 1 j S (t) 0 Λ (31) from which the stronger inequality follows W[0,T ren ] F (ρ(t )) F (ρ(0)) (32) Equality is obtained for quasi-static transformations. In fact in such a case the integral in (31) can be made as small as we want. The idea of renormalized work was introduced in Y. Oono, M. Paniconi, Prog. Theor. Phys. Suppl. 130, 29 (1998).

34 An alternative renormalization C. Maes, K. Netocny, arxiv: One may ask whether there exist alternative renormalizations of the total work. For instance, in a recent work, Maes and Netocny considered the topic of a renormalized Clausius inequality in the context of a single Brownian particle in a time dependent environment. To compare their approach to the present one, consider N independent diffusions in the thermodynamic limit N. Each diffusion solves the Langevin equation Ẋ = E(t, X) + 2 ẇ, where E is a time dependent vector field and ẇ denotes white noise. The corresponding stationary measure with E frozen at time t is denoted by exp{ v(t, x)}.

35 The scheme discussed here can be now applied. The transport coefficients are D = 1 and χ(ρ) = ρ. We subtract the energy dissipated by J A (t, ρ) = ρ [ E(t, x) + v(t, x) ]. The renormalization introduced in Maes and Netocny is instead obtained by introducing a potential field such that the corresponding stationary state has minimal entropy production. Namely, they write E = f U and subtract from the energy exchanged the space-time integral of J φ t 2 /ρ where J φ t = ρ(f φ) ρ and φ = φ(t, x; ρ) is chosen so that J φ t = 0. While the two renormalization schemes are different, both satisfy a Clausius inequality with F (ρ) = dx ρ log ρ. Observe that in this case of independent particles our renormalization is local while the dependence of J φ t on ρ is nonlocal.

36 Analysis of the energy balance L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim, paper in preparation The idea is to expand the two sides of the identity T 0 { } dt dσ(x) λ(t, x) j(t, x) ˆn(x) + dx j(t, x) E(t, x) Λ Λ =F(ρ(T )) F (ρ(0)) + T 0 dt Λ dx j(t) χ(u(t)) 1 j(t) in powers of 1/T

37 For simplicity we here restrict the discussion to one space dimension. Let E(s) and λ(s) with s [0, 1] be a protocol. The slow transformation is then realized by { E τ (t) = E (t/τ), λ τ t [0, τ]. (t) = λ (t/τ), Let also ρ τ (t) and j τ (t), 0 t τ, be the solution to t ρ τ + J(t/τ, ρ τ (t)) = 0, j τ (t) = J(t/τ, ρ τ (t)) f (ρ τ (t)) Λ = λ τ (t) ρ τ (0) = ρ(0) (33) where we recall that J(t, ρ) = D(ρ) ρ + χ(ρ)e (t), and ρ(0) is the unique stationary solution of the hydrodynamics with external field E(0) and chemical potential λ(0).

38 Along the trasformation (ρ τ, j τ ) the energy balance can be written 1 1 τ ds dx j τ (τ s) E(s) τ ds dσ λ(s)j τ (τ s) ˆn 0 Λ 0 Λ = F ( ρ τ (τ) ) F ( ρ(0) ) (34) 1 + τ ds dxj ( s, ρ τ (τs) ) χ ( ρ τ (τs) ) 1 ( J s, ρ τ (τs) ) 0 Λ

39 We now expand in 1/τ ρ τ (τs) = ρ(s) + 1 τ r(s) + O( 1 τ 2 ), s [0, 1] where ρ(s) is the stationary solution to the hydrodynamic equation having external field E(s) and chemical potential λ(s). j τ (τs) = J(s, ρ(s)) + 1 τ g(s) + O( 1 τ 2 ). (35) r solves s ρ(s) + g(s) = 0 g(s) = D( ρ(s)) r(s) r(s) [ D ( ρ(s)) ρ(s) + χ ( ρ(s))e(s) ] r(s, x) = 0, x Λ (36) which has the form of a Poisson equation for r(s).

40 We obtain to order 0 in 1/τ F ( ρ(1) ) F ( ρ(0) ) 1 1 = ds dx E(s) g(s) ds dσ λ(s)g(s) n 0 Λ 0 Λ 1 ds dx r(s) χ ( ρ(s) ) χ 2( ρ(s) )J 2( s, ρ(s) ). 0 Λ (37)

41 Equilibrium Equilibrium Consider a transformation from an equilibrium state to another one in a one-dimensional setting assuming E = 0 and λ(0) = λ 0, λ( ) = λ 1. Eq. (37) takes the simpler form F ( ρ(1) ) F ( ρ(0) ) 1 = ds dσ λ(s)g(s) n 0 Λ 1 ds dx r(s) χ ( ρ(s) ) (38) χ 2( ρ(s) )J 2( s, ρ(s) ). 0 Λ If the transformation goes through equilibrium intermediate states the last term does not contribute. However it does contribute if the intermediate states are nonequilibrium stationary states. Therefore for a given F one can compare, for a given protocol, the work provided by the reservoirs with the work provided by the dynamics of the system.

42 For large finite τ a direct calculation shows that where and B = W ren [0,τ] = F + 1 τ B + O( 1 τ 2 ). (39) 1 0 C 1 s ds dx χ( ρ(s)) [ Cs 1 r(s) ] 2 > 0 (40) Λ (x, y) = δ2 V λ(s),e(s) ( ρ(s)). δρ(x)δρ(y) We observe that B depends on the protocol but it has a strictly positive lower bound so we can select the best protocol by minimizing B.

43 Dissipation The infinitesimal version of the identity (26) gives the istantaneous energy balance which reads Ẇ = dx [ f (ρ) ρ + j χ(ρ) 1 j ] (41) Λ where Ẇ is the power injected by the reservoirs and external field in the system. Accordingly, f (ρ) ρ represents the rate of change of the free energy while j χ(ρ) 1 j is the dissipated power.

44 In general the dissipation is not minimal in the stationary state. As a simple example let us consider the case of independent particles. the minimizer of the second term on the right with the prescribed boundary conditions ρ(0) = ρ 0, ρ(l) = ρ 1 is ˆρ(x) = [ ρ0 (1 x/l) + ρ 1 x/l ] 2 while the stationary profile is ρ(x) = ρ 0 (1 x/l) + ρ 1 x/l. Observe that, in accordance with the Prigogine principle, ρ ˆρ = O([(ρ 1 ρ 0 )/L] 2 ).

45 The quasi-potential as relative entropy L. Bertini, D. Gabrielli, G. Jona-Lasinio, C. Landim, (2012), J. Stat. Phys. 149, 773 (2012) The relative entropy S(ν µ) of the probability ν with respect to µ is defined by S(ν µ) = dµ dν dν log dµ dµ (42) We discuss the case of stochastic lattice gases. If Λ R d is the macroscopic volume and Λ ɛ the corresponding subset of the lattice with spacing ɛ, the number of sites in Λ ɛ is approximately ɛ d Λ. Given (λ 0, E 0 ) and (λ 1, E 1 ), we claim that lim ɛ 0 ɛd S ( µ λ 0,E 0 Λ ɛ µ λ 1,E 1 ) Λ ɛ = Vλ1,E 1 ( ρ 0 ), (43) where ρ 0 is the stationary profile corresponding to (λ 0, E 0 ).

46 In view of the definition (42) of the relative entropy we have that ɛ d S ( µ λ 0,E 0 Λ ɛ µ λ 1,E 1 ) Λ ɛ = ɛ d η µ λ 0,E 0 Λ ɛ (η) log µλ0,e0 Λ ɛ (η) µ λ 1,E 1 Λ ɛ (η). By the large deviation formula we then get ɛ d S ( µ λ 0,E 0 Λ ɛ µ λ 1,E 1 ) Λ ɛ ɛ d β η µ λ 0,E 0 Λ ɛ (η) [ V λ1,e 1 (ρ ɛ (η)) V λ0,e 0 (ρ ɛ (η)) ] β [ V λ1,e 1 ( ρ 0 ) V λ0,e 0 ( ρ 0 ) ] = β V λ1,e 1 ( ρ 0 ), where ρ ɛ (η) denotes the density profile associated to the microscopic configuration η. In the final step we used the law of large numbers for the microscopic density profile under the probability µ λ 0,E 0 Λ ɛ.

47 False counterexamples Let, for instance, µ β ɛ be the Gibbs measure for a one-dimensional Ising model at zero magnetic field and inverse temperatures β 0 and β 1 on a ring with ɛ 1 sites. The magnetization satisfies a large deviation formula and its typical value is zero for both ensembles so that the right hand side of (43) vanishes. On the other hand, by a direct computation, for β 0 β 1, lim ɛ ɛs(µ β 0 ɛ µ β 1 ɛ ) > 0. However this example does not contradict (43) as we are comparing two ensembles in which we varied the temperature and not the magnetic field. In this example, the correct formulation of (43) would have been in terms of the large deviation function for the energy, that is the extensive variable conjugated the the intensive parameter that has been changed.

48 4. SUMMARY

49 The theory presented provides a selfcontained macroscopic description of diffusive systems out of equilibrium. The approach has been largely inspired by microscopic models with a stochastic dynamics where explicit calculations and mathematical proofs are possible. Lattice gases and simplified models of heat transport seem to play a role akin to the Ising model in equilibrium statistical mechanics.

50 We have introduced a natural notion of renormalized work which satisfies a Clausius inequality and we have shown that quasi static transformations are optimal with respect to it. An expansion in the inverse of the duration of a thermodynamic transformation allows a separation of the contribution of the environment to the change of the free energy from the work done by the system. We have established a connection between the large deviation functional and the work involved in certain transformations providing in this way a thermodynamic definition of the quasi potential. We have shown that the large deviation functional coincides with the relative Shannon entropy of the final state with respect to the initial state.