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Introduction

The fluctuation-dissipation relation (FDR) is a principle of statistical mechanics that relates spontaneous fluctuations (say the random current in a piece of metal) and response to external fields (say the current created by an external electric field). This principle takes various forms depending on the setting (equilibrium or not, linear or not, etc.). It has known relatively recent developments (by Green-Kubo,  Evans-Cohen-Morris,…) and  is currently much used to estimate characteristics like susceptibilities from numerical simulations. It has been considered in other fields like dynamical systems (see, e.g., this preprint) and probability theory (see this other one).

Equilibrium states

The simplest results assume equilibrium.

Fluctuations and susceptibility

According to the canonical formalism, for a system at temperature $1/\beta$,  if $\mu(S)$ is the volume of microstates whose energy belong to $S$, the probability of having energy at most $x$ is: $P_\beta(E where $\psi$ is a normalizing function. It follows at least formally that the averages under $P_\beta$ satisfy:

$_\beta = \psi'(\beta)$ and $_\beta-_\beta^2 = -\psi''(\beta)$

Hence the variance of the spontaneous fluctuations of the average of the energy is equal to its $\beta$-derivative  (the heat capacity).

This holds more generally if one replaces $(E,\beta)$ by any pair of conjugate thermodynamical variables $(X,h)$ provided that one can assume that the perturbation to the equilibrium state density is linear: $f(P,Q)=f_0(P,Q)(1-\lambda\beta(A-_0))$ where $\lambda A(P,Q)$ is the conjugating term in the Hamiltonian and $<\dot>_0$ is the average when $\lambda=0$. One then gets: $(-_0 )/\lambda = \beta(_0-_0_0)$. The susceptibility is therefore again the variance of equilibrium fluctuations  if $\lambda A\equiv \lambda B$.

Stationary states

More sophisticated approaches consider the Hamiltonian dynamics starting from the equilibriu

Kubo’s formula

It deals with a system with Hamiltonian $H(P,Q)=H_0(P,Q)+F(t)A(P,Q)$, the last term representing the external field. If $B(P,Q)$ is an arbitrary test function, the change in its average at time $t$ is given by:$<\Delta B(t)> =\int_0^t ds R(t-s)F(s)$ with the response function $R(t) = \beta < - \{H_0, A\} . B\circ \Phi_0^t) >_0$ where $\{\cdot,\cdot\}$ is the Poisson bracket and $\Phi_0^t$ is the flow defined by $H_0$ and $<\cdot >_0$ denotes the average with respect to the equilibrium for Hamiltonian $H_0$.

If one introduces the dissipative current is $J(P,Q):=\sum_j ((-\partial A/\partial q_j)F^0_j-(\partial A/\partial p_j)p_j/m)$ where the sum is over the particles and $F^0_j=-\partial H_0/\partial q_j$, the response function can be written $R(t)= -\beta < J(0)B(t) >_0$. Let us apply it to a uniform, constant force along the $x$-direction starting at $t=0$: $A(P,Q)=-F\sum_j q^x_j$, $F(t)=H(t)F$. It follows: $J(P,Q)=-(F/m)\sum_j p^x_j$ and $<\Delta V^x(t) >= \beta F\int_0^t ds \sum_j < v^x_j(t_0) v^x_j(t-s) >_0$. The transport coefficient $<\Delta V^x(t) >/F$  is thus proportional to the fluctuations  at equilibrium (as measured by the self-correlation of the $x$-velocities).

Linear response in stochastic dynamics

This can be analyzed using Langevin dynamics: $\dot X = -\gamma A'(X)-F(t)+\sqrt{2\gamma T}\eta$ where $A$ is a function, $\gamma, T$ are positive constants, $\eta$ is a white noise. The response function is: $R$ suc that: $X(t) = \int_0^t R(t,s) F(s) \, ds$ can be computed: $R(t,s)= (\gamma T)^{-1}H(t-s)\partial C(t,s)/\partial s$ where $C(t,s):=_0$ (the average when $f\equiv0$).