# Can an Ergodic dynamical system approach equilibrium?

An ergodic dynamical system $(\Omega,\phi^t,\mu)$ is such that the time average $\bar{f}$ of every function $f\in L_1(\Omega,\mu)$ equal the space average $\langle f \rangle_\mu$, i.e. the system densely cover all the phase space ($\mu$-almost everywhere). Another equal condition of ergodicity is that the only invariant sets ($\phi^t(B)=B$) are the trivial ones (no way of partitioning the phase space), or looking at the $L_1$ space, the only invariant functions ($f\circ\phi^t=f$) are the constant functions. Moreover, we have a stronger property than ergodicity, namely mixing that implies the former. A system is said to be mixing if $$\mu(\phi^{-t}(A)\cap B)\rightarrow\mu(A)\mu(B),\qquad\text{as }t\rightarrow\infty.$$ At the end we have the recurrence, i.e. the system pass through all the points of the phase space infinitely many times. My question arise because in my mind for a system that approaches the equilibrium, there exist a time $T>0$ such that for all $t>T$ the system will spend its future time in a smaller region of the phase space, i.e. there exist a partition of $\Omega$ and then the system cannot be ergodic.

• I think one often considers the phase space given that the system is in equilibrium and tries to see if this equilibrium system is ergodic, or am I missing something here?
– Danu
Commented Aug 29, 2014 at 14:02
• Ok, but for definition an ergodic system cannot evolves towards equilibrium. Or i am missing something? Commented Aug 29, 2014 at 14:09
• I think one can technically say that an ergodic system will not be capable of forever maintaining equilibrium at all instances, but I think this may be irrelevant because of averaging (I really don't know!)... Also, this question is related, please read the last section of my answer.
– Danu
Commented Aug 29, 2014 at 14:11
• I am not sure I understand your concern. I assume that your dynamical system is measure-preserving (as is the relevant setting for application to stat. mech.). In that case, it is clear that the system does not "spend its future time in a smaller region of the phase space", at least not as measured with $\mu$. Maybe you'd enjoy reading this very nice paper, which although not discussing specifically the issue you have, might well clarify things for you. Commented Aug 29, 2014 at 14:29
• What is your definition of equilibrium? In physics an ergodic system is only "in equilibrium" during most of its dynamic evolution. Indeed, the recurrence theorem guarantees that all systems that start from a very special state (i.e. far from equilibrium) will, eventually, return arbitrarily close to that special state, again. Commented Aug 30, 2014 at 5:27

Maybe i found the source of my concern, the whole space phase have invariant measure so it's measure never change during the evolution and so i think i have a misleading idea of equilibrium approach, indeed it cannot be ''the system will spend it's future time in a set $A\subset\Omega$''.