Harmonic oscillator in Classical Statistical Mechanics

1) The total Hamiltonian and full phase space

We split the world into a small system $S$ and a huge bath $B$. The total Hamiltonian is

where, as an example, for a 1-D harmonic oscillator

The full microstate is a single point $\mathrm{\Gamma }=(q,p,\{q_i,p_i\})$ in the huge phase space (dimension $2(1+N)$ if $N$ bath particles).

Every coordinate $x_k$ in $\mathrm{\Gamma }$ evolves deterministically (Hamiltonian equations):

and if we consider an ensemble (infinitely many copies) we describe it by the phase-space density ${\rho }_{\text{tot}}(\mathrm{\Gamma },t)$. Conservation of probability gives the Liouville equation:

which is a continuity equation.
Liouville's theorem implies the flow is incompressible in phase space: volumes are preserved. If ${\rho }_{\text{tot}}(\mathrm{\Gamma },0)=\delta (\mathrm{\Gamma }-{\mathrm{\Gamma }}_0)$ it stays a delta moving along the trajectory $\mathrm{\Gamma }(t)$. I intuitively understand this, but it needs clarification.

2) We only care about the small system — integrate out the bath

We define the reduced density for the subsystem

where $d{\mathrm{\Gamma }}_B$ denotes integration over all bath degrees of freedom. This is the precise mathematical statement of “I don’t know (and don’t want) the bath”. It is a marginal distribution.

If you integrate the Liouville equation over ${\mathrm{\Gamma }}_B$ you get an evolution equation for ${\rho }_S$ that depends on bath-system correlations. Symbolically,

That extra term prevents a closed, simple deterministic equation for ${\rho }_S$ unless you make further approximations.

3) Extra physical assumptions allow closure

To replace the complicated coupling term with something manageable we usually make several assumptions (not listed here) in such a way that we obtain a simple equation (master equation) for ${\rho }_S$. The general structure is:

where $\mathcal{D}$ is a dissipative (non-Hamiltonian) operator that encodes friction and noise.

Indeed, we sometimes assume some equilibrium (like thermal equilibrium at temperature $T$) in such a way that ${\rho }_S$ gets time-independent. For example, in the case of the harmonic oscillator the thermal equilibrium hypothesis lead to the stationary Boltzmann distribution

4) Intuition for why irreversibility appears

• The total system evolves reversibly, but we deliberately discard information about the bath (we integrate it out).
• The discarded correlations reappear in $\mathcal{D}[{\rho }_S]$ as stochastic noise and friction.
• Even worse, with extra assumptions we even get a time-independent ${\rho }_S$, so we have completely discarded the recovery of the initial setup.

5) Average energy

For a general thermal equilibrium system we have this.

In the particular case of the harmonic oscillator in thermal equilibrium we have, due to the equipartition theorem, that

6) Entropy

Two different entropy notions are useful and often confused:

• Gibbs (fine-grained) entropy. In general,

Under Liouville evolution $S_G$ is constant in time (the flow is incompressible).
But for a harmonic oscillator in thermal equilibrium, the entropy for the "small system" can be seen to satisfy

(see entropy#Example). Then,

We can take $C=-\ln T_0$ and then

This can be worked out (see "What is Entropy" by J. Baez) to

• Boltzmann / coarse-grained entropy: partition phase space into macrostates and define $S_B=k_B\ln \mathrm{\Omega }$ where $\mathrm{\Omega }$ is the volume of the macrostate. ?????????????
  Thus irreversibility appears because we describe the system with a coarse, reduced description (lose microscopic correlations), not because the microscopic laws are irreversible.