Gibbs measures

 

These are the notes that I’m taking for myself in order to learn some statistical mechanics.

The main reference is: Anton Bovier, Lecture notes on Gibbs measures and phase transitions (Bonn University)

In thermodynamics one has:

E = E_{mech} + E_{chem} + E_{thermal}

where E_{mech}, E_{chem}, E_{thermal} are respectively the mechanical, chemical and thermal components of the energy.

E_{thermal} is also denoted by Q,

dE_{mech} = - p dV

where p is the presure and V is the volume

d E_{chem} = \mu dN

where N is the number of  molecules, and \mu is the average chemical potential per molecule

dQ = T dS

where T is the temperature and S is the volume

Thus we have, for a closed system:

0 = dE = -pdV + \mu dN + TdS

We will view the total energy E as a function of three extensive variables V, N, S: E = E(V,N,S). Then we have

p = - \partial E / \partial V \mu = \partial E / \partial N T = \partial E / \partial S

p, \mu, T are called intensive variables. The last three equations are called equations of state.

Instead of viewing E as a function of three variables V,N,S one can also view it as a function of another set of three variables, e.g. p,N,T, etc. We will assume that E is a convex function, so that change is possible. (Legendre transformation). Problem: it can’t be convex ?! Howevern can have a nondegenerate Hessian ?!

Intorduce a function G (of three variables p,T,N) such that

dG = Vdp - SdT + \mu dN = d(Vp - ST + E)

So in fact

G = Vp - ST + E. This sunction is called the Gibbs free energy. Some other useful functions are:

Helmholtz free energy: F = E - TS

Enthalpy: H = E + pV

In some case E is linear in V: p = \partial E/\partial V is a constant (then we can’t compute V as a function of p). This situation is called a first order phase transition. (Example: condensation of vapor into water).

1-dimensional model:

N particle, volume V, pressure f, each particle has moment p_i

Total energy function (conserved quantity): H = fV + \sum p_i^2/2 is a constant, V \leq V_{max} = H/f

Assume equi-distribution of all possible positions and momenta x_i,p_i having a fixed E, the level V which has the highest probabality will be (assuming N very large): V = 2/3 . E/f. (This is the level that one will observe in reality).

The probability formula for v \in dv is:

P(v \in dv) = \frac{dv \exp(Ns)}{\int_0^{v_{max}} \exp(Ns)}

where

s = \ln v + 1/2 \ln (2m(u-fv)), u = H/N, v = V/N, e = E/N, E = \sum p_i^2/2.

Equilibrium state:

v = 2/3 u/f, T = 2e, {\partial e \over \partial s} = T.

Everything works very nicely in this 1-dimensional model ! This model also indicates why the entropy is a log function, and what does it mean to be a micro-state.

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