This text provides a readable introduction to Markov processes,

including both Langevin and Fokker-Planck equations, from the

standpoint of typical physical examples. So why review an ancient

book on statistical physics in an age when we're already into

complexity? After all, every econophysicist already knows the

contents of this book. Right? Well, we as physicists are sometimes

good at criticizing the economists for mistakes about equilbrium that

they propagate in their literature, but we generally ignore the same

misconceptions in our own statistical physics literature, and this

book apparently treats stochastic processes from a standpoint that

assumes that stochastic processes are always near equilibrium. Where

does the mistake creep into this otherwise fine text?

There are two mistakes on pages 65-68. The discussion is based on the

stochatic differential equation (sde)

dx=-R(x)dt+D(x,t)^1/2dB(t)

where B(t) is a Wiener process. First, it is claimed that the random force

D(x,t)^1/2dB

is Gaussian with a white spectrum. In general, though, the random

force cannot be stationary unless D is independent of x. The unstated

assumption of the text seems to be that the random force is always

stationary, so that with R(x)
When the diffusion coefficient depends on x (or more generally on

(x,t)) then there can be no approach to equilibrium for the case of

unbounded x, even with R
finance theory so vividly shows. In the lognormal model, e.g., there

exists an equilibrium solution that is easily calculated in closed

algebraic form, but that solution is not reached by time-dependent

solutions of the Fokker-Planck equation. More generally, even if an

equilibrium solution of the Fokker-Planck equation for a variable

diffusion coefficient (or 'local volatility') D(x) 'exists', it

cannot be reached dynamically when the random force is nonstationary.

Demonstrationing that an equilibrium solution 'exists' is meaningless

are useless if the dynamics can't approach that solution. We know now

that the empirical finance distribution requires an (x,t)-dependent

diffusion coefficient, and that the empirical distribution does not

approach equilibrium as time increases. Finance theorists often

massage their data by 'transforming' to a stationary distribution.

This is a terrribly misleading approximation. In general, stationary

distributions do not exist empirically, especially not where

complexity comes into the picture.

A final point of interest: here's how to test empirically for the

stabilizing effect of Adam Smith's Invisible Hand: check to see if

the economic data are stationary. If not, then the 'Invisible Hand is

unreliable. The problem you'll face is that most economic data are

too poor to test reliably for stationarity, or for anything else, for

that matter (the data are generally too easy to fit). But maybe

Bertrand Roehner can help by pointing us to the better data in

nonfinancial economics.

1

vote
## Nonequilibrium Statistical Mechanics

Ryugo Kubo

posted on 30 June 2003

reviewed by Joe McCauley

## Discussion

Essa análise é muito bem feita