# What is the intuition behind the statement that non-equilibrium systems with static disorder are self-averaging?

In this paper(1) by C. De Dominics, he makes the argument that in a dynamical statistical mechanics system, one doesn't need to apply the replica trick and can directly disorder average the generating function $$\bar Z[j] = \langle Z[j] \rangle_{dis}$$ leading to a two-time effective action (j= source).

He doesn't really explain why this is ok to do, at least as far as I understood his vague arguments:

Indeed, the trace over random degrees of freedom can then be performed ab initio leaving a Lagrangian form where all random degrees of freedom have been eliminated by integration. The removal of closed loops, which is achieved in the replica method by giving them vanishing weight n, comes about in the present method by the impossibility to build closed loops since they cannot be constructed from retarded propagators alone.

All I understand is that somehow the fact that the propagator $$G(t) \propto \theta(t)$$ leads to a self averaged generating function? He never actually demonstrates any of this and just proceeds to some calculations. Can someone help me understand the statement?

Source:

(1) C. De DominicsPhysical Review B Volume 18, Number 9, 1 November 1978

"Dynamics as a substitute for replicas in systems with quenched random impurities"

• You should probably at least cite the paper properly (title, author, journal, year). Without access it's not even possible to know which paper we're talking about. – valerio Jan 26 at 0:00
• No I believe your question is quite different. This is not a question of annealed vs. quenched. I am asking why in a non-equilibrium system with quenched (totally non-dynamical) disorder, one can directly average the partition function. This is a system evolving under some fixed Markov process who's transition matrix is drawn from a distribution. – Adam B Jan 26 at 0:52

I've seen this claim echoed in other references, e.g. in Jordan Rammer's book Quantum Field Theory of Non-Equilibrium States pg. 455 below equation 12.33, however I haven't found a satisfying discussion. I would love to see a rigorous discussion of this.

My best guess based on the references is the following. Take some statistical field theory of a dynamic process coupled to a static random variable with the sourced partition function $$Z[j]$$. Expectation values are computed as $$\langle A (t) \rangle = \frac{A\left(\tfrac{\delta \;}{\delta j(t)}\right)Z[j]}{Z[j]}\big|_{j\rightarrow0}= \frac{\int D[\phi,\phi^*]A\left(\phi(t),\phi^*(t)\right)e^{-S_{sys}}}{Z[0]}$$ Then the disorder average $$[\cdots ]_{dis}$$ of this is given by $$[\langle A (t) \rangle ]_{dis}=\left[ \frac{\int D[\phi,\phi^*]A\left(\phi(t),\phi^*(t)\right)e^{-S_{sys}}}{Z[0]}\right]_{dis}$$

I suppose that $$Z[0]^{-1}$$ typically interferes with the averages in a complex way; however in a dynamical system where $$Z[0]=1$$, it is necessarily independent of the disorder so that the result is simply given by the disorder average over $$Z$$

$$[\langle A (t) \rangle ]_{dis}=\left[ \int D[\phi,\phi^*]A\left(\phi(t),\phi^*(t)\right)e^{-S_{sys}}\right]_{dis}=A\left(\tfrac{\delta \;}{\delta j(t)}\right)\left[Z[j]\right]_{dis}\big|_{j\rightarrow0}$$

This pushes the question to the meaning of the normalization $$Z[0]=1$$ (Lots of sketchy stuff happened here with interchanging limits and integrals, so take this with a grain of salt.) I'm not well versed in the classical field theory methods, but the normalization seems to arise from conservation of probability -- e.g. the partition function is itself a literal representation of the probability distribution function and always evaluates to unity, in contrast to thermodynamic and quantum systems where we know it evaluates to the distribution as a function of parameters.

Hopefully someone can improve on this response! I've decided to start a bounty because I think the proof/rigorous discussion would be of interest to a large subset of physicists.

I think the response by user kapaw is largely correct; I will try to simply expand on it in relation to the formalism.

Indeed, the replica trick can be circumvented provided we ensure that the partition function has value unity when there are no external sources. Then, once all derivatives are taken, $$Z[0]$$ drops out of the expressions and in particular, the quantity $$Z[0] = 1$$ is independent of the realization of the quenched disorder.

The paper seems to be formulating the problem in terms of the MSR method, which maps a classical stochastic equation of motion onto an equivalent functional integral (see, e.g. Kamenev's book (1) or Altland and Simons' book (2) for more details).

This integral is effectively the classical version of the Keldysh path integral for quantum non-equilibrium systems. The important construction here is that the time-evolution is performed on two contours, from $$t=-\infty$$ to $$t= +\infty$$ and then immediately back again to $$t=-\infty$$ from $$t=+\infty$$. This ensures that any field which takes the same values on both contours will cancel, effectively subtracting off all disconnected diagrams. This formulation is usually re-written to produce a diagrammatic method for computing the retarded, advanced, and Keldysh (abbreviated RAK) correlation functions perturbatively.

In particular, if I recall correctly, the cancellation of the vacuum fluctuations and the guarantee that $$Z[0]=1$$ is essentially due to a symmetry of the non-equilibrium action which emerges in the RAK basis. Thus, it is effectively proven by a Ward identity of the theory. That about exhausts my knowledge of the subject, but hopefully that contributes to your remaining open question!

Sources:

(1) Kamenev, A. "Field Theory of Non-Equilibrium Systems," Cambridge University Press (2011)

(2) Altland, A. and Simons, B., "Condensed Matter Field Theory," Cambridge University Press (2010)