Entropy and the principle of least action

Question

Is there any link between the law of maximum entropy and the principle of least action. Is it possible to derive one from the other?

Answer 1

Yes, there is a link, both are examples of extremum principles. And yes it is possible to derive the principle of least action from the law of maximum entropy. The derivation is lengthy and I will only sketch the main steps needed:

1. We start from the law of maximum entropy $dS/dt \geq 0$. As we know this law is only valid for isolated systems [i]. For dissipative systems $dS/dt > 0$, the evolution is irreversible and cannot be described by an action principle. We must consider non-dissipative systems, for which $dS/dt = 0$. This is correct because the action principles are rigorously restricted [ii] to Nondissipative Systems.

2. From the phase space structure we can show that the phase space state $\rho$ satisfies the equation $d\rho/dt = \partial\rho/\partial t - \mathcal{L}\rho$, where $\mathcal{L}$ is the Liouvillian.

3. From the constancy of entropy (1), we can derive the Liouville theorem $d\rho/dt=0$, using the Gibbs relation $S=S(\rho)$. This implies that the general equation of motion (2) reduces to the Liouville equation $\partial\rho/\partial t = \mathcal{L}\rho$. Effectively, this equation is not dissipative and conserves entropy.

4. For a mechanical system in a pure state, the phase space state is given by the well-known product of Dirac deltas; substituting this $\rho_\mathrm{pure}$ on the Liouville equation, the equation reduces to the Hamilton equations of motion: $dq/dt = \partial H / \partial p$ and $dp/dt = -\partial H / \partial q$.

5. Using the Hamilton Jacobi method, the Hamilton equations of motion can be written again as a single equation: the Hamilton Jacobi equation $H + \partial A / \partial t = 0$, where $A$ is the action [iii].

6. It can be shown that the Hamilton Jacobi equation "is an equivalent expression of an integral minimization problem such as Hamilton's principle", and Hamilton's principle is just the Hamiltonian version of the principle of least action. In other words, solving the Hamilton Jacobi equation one obtains the action $A$ and this automatically satisfies the principle of least action $\delta A=0$.

7. Other versions of the principle of least action can be obtained from here. For instance, the Lagrangian version of the principle can be obtained using a Legendre transformation for deriving the Lagrangian $L = pv - H$. In this case, the action is given by $A=\int L dt$.

[i] For non-isolated system entropy can increase, decrease or remain constant.

[ii] As reported in the Scholarpedia link there are some few special dissipative systems which can be described by an action principle. Those are open systems for which the production of entropy is compensated by an external flow of entropy to give a zero total variation $dS=d_iS+d_eS=0$. Moreover, the action principle only describes the average behaviour of these systems, but not the thermal fluctuations.

[iii] References on mechanics usually denote the action by $S$, but here it would be confused with entropy.

Answer 2

"Entropy" and "action" are two entirely different concepts. The first relates to a coarse-grained statistical description of a physical system at macroscopic scales, the latter to the underlying deterministic microscopic dynamics exhibited by the system.

Also note that: 1) the second law of thermodynamics tells us entropy doesn't decrease, it need not increase and certainly can attain non-maximal values, and 2) action is stationary and not necessarily minimal or maximal. Hence, when considered as fundamental physics laws, both 'maximal entropy' and the 'principle of least action' are misnomers.

Zooming in to the core of your question: 'entropy increase' and 'action stationarity' are unrelated, and even incompatible. One certainly can not be derived from the other. This is for the simple reason that 'action stationarity' describes a reversible physics, while 'entropy increase' presents us with an irreversible picture of evolution of physical systems. The difference, again, is in microscopic versus macroscopic.

As an analogy, think about two statements one can make about the physics of pool billiard. The first being that the balls collide according to Newtons laws which can be expressed by stating that the detailed balls trajectories render a quantity called 'action' stationary. The second being the coarse-grained statistical statement that as long as balls aren't pocketed yet, the mean distance between the balls doesn't decrease. Both statements are unrelated and apply to a description of pool billiard at different levels.

Answer 3

As Joannes says, the two principles belong to two different theories:

• the least action principle is a principle about the (conservative) laws of motion and a proposition about the paths actually followed by the degrees of freedom of mechanical bodies

• the maximum entropy principle refers either to thermodynamics to figure out in which direction will a transformation occur or more generally to Bayesian inference theory and is rarely concerned with dynamics

They are thus quite different beasts conceptually and, although there may be some overlaps between the two in some instances, I would not recommend, as a matter of principle, to think that these two things are closely related as it is not the case.

As for "deriving" one from the other, at the very least we should agree first about what are the axioms we start from.

For instance, some versions of Crook's fluctuation theorem enable one to get something very close to a maximum entropy principle (on average) "from" simply hamiltonian mechanics (which I would put on the same footing as the least action principle for now) but it is clear that the theorem relies a lot on probability theory (and possibly on Bayesian inference) which, in my view at least, is outside the scope of the least action principle alone.

I think that's an important shift to notice even when invoking Liouville's theorem which is a theorem that makes propositions about probability densities and not about trajectories.

Answer 4

This old russian paper seemed to have discussed the problem without going much into non-equilibrium statistical mechanics. I haven't read the entire thing. Soviet Physics Journal May 1991, Volume 34, Issue 5, pp 426-431. Moslov is the author. Check out their other papers too.