8.1: Equilibrium, Nonequilibrium and Local Equilibrium
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Classical equilibrium statistical mechanics is described by the full N-body distribution, f0(x†1,…,x†N;p†1,…,p†N)={Z−1N⋅1N!e−βˆH∗N(p,x)OCEΞ−1⋅1N!eβμNe−βˆH∗N(p,x)GCE .
Nonequilibrium statistical mechanics seeks to describe thermodynamic systems which are out of equilibrium, meaning that the distribution function is not given by the Boltzmann distribution above. For a general nonequilibrium setting, it is hopeless to make progress – we’d have to integrate the equations of motion for all the constituent particles. However, typically we are concerned with situations where external forces or constraints are imposed over some macroscopic scale. Examples would include the imposition of a voltage drop across a metal, or a temperature differential across any thermodynamic sample. In such cases, scattering at microscopic length and time scales described by the mean free path ℓ and the collision time τ work to establish local equilibrium throughout the system. A local equilibrium is a state described by a space and time varying temperature T(r,t) and chemical potential μ(r,t). As we will see, the Boltzmann distribution with T=T(r,t) and μ=μ(r,t) will not be a solution to the evolution equation governing the distribution function. Rather, the distribution for systems slightly out of equilibrium will be of the form f=f0+δf, where f0 describes a state of local equilibrium.
We will mainly be interested in the one-body distribution f(r,p;t)=N∑i=1⟨δ(x∗i(t)−r)δ(p∗i(t)−p)⟩=N∫N∏i=2ddx∗iddp∗if(r,x∗2,…,x∗N;p,p∗2,…,p∗N;t) .
In the GCE, we sum the RHS above over N. Assuming v=0 so that there is no one-body potential to break translational symmetry, the equilibrium distribution is time-independent and space-independent: f0(r,p)=n(2πmkBT)−3/2e−p2/2mk∗BT ,
Thermodynamics says that dq=Tds=dε−μdn ,
When the individual particles are not point particles, they possess angular momentum as well as linear momentum. Following Lifshitz and Pitaevskii, we abbreviate Γ=(p,L) for these two variables for the case of diatomic molecules, and Γ=(p,L,ˆn⋅L) in the case of spherical top molecules, where ˆn is the symmetry axis of the top. We then have, in d=3 dimensions, dΓ={d3ppoint particlesd3pLdLdΩ∗Ldiatomic moleculesd3pL2dLdΩ∗Ldcosϑsymmetric tops ,