Statistical mechanics and thermodynamics

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summarized by Lior Yosub and Liana Diesendruck

Contents

[edit] Main topics of Statistical Mechanics and Thermodynamics 2 BGU course:

  • Phase transitions.
  • Kinetic theory.
  • Quantum gases.

[edit] Review

Let's begin with a small review of Thermodynamics and Statistical Mechanics I. We defined 3 types of ensemble:

[edit] Micro-Canonical Ensemble

In this ensemble we set three macroscopic parameters - Energy (U), Volume (V) and Number of Particles (N). The probability of a non-accessible state is 0 and the of an accessible state is \frac{1}{g}, where g is the total number of accessible states, called multiplicity function or degeneracy

\begin{matrix} &P(accessible)=\frac{1}{g},\ where\ g=\sum_{accessible}1, \\ &P(non-accessible)=0 \end{matrix}

We define the entropy as σ = lng, and in a closed system it may only rise. This is the second law of Thermodynamics and will guide us through this course. The change in the energy of a system is dU = δQ + δW, where δQ is the change in the heat (δQ = τdσ) and δW is the change in work. The definition of δW differ from system to system, but we usually consider mechanical work dW = − pdV. Now we can write the explicit change of energy as:

dU = τdσ − pdV

Now we should write the same formula from a different perspective: in closed system the energy and the volume are fixed, thus we define enthropy as function of these quantities: \sigma\left(U,V\right)

d\sigma=\frac{dU}{\tau}+\frac{p}{\tau}dV

from here it is easy to see the following relations:

\frac{1}{\tau}=\left(\frac{d\sigma}{dU}\right)_{V,N} \frac{p}{\tau}=\left(\frac{d\sigma}{dV}\right)_{U,V}

[edit] Canonical Ensemble

In this ensemble we attach our system to a heat reservoir in order to fix the temperature of our system. We assume that the heat reservoir is extremely large and it's temperature is not affected by the heat transferred from (or to) our system. In addition to setting the temperature τ we also set constant the Volume (V) and the Number of Particles (N) and release the hold on the Energy (U). The probability of finding the system in particular state is defined as:

P(\left\{s_{i}\right\})=\frac{1}{Z}e^{-\frac{\epsilon_{s}}{\tau}}

where Z=\sum_{\{s\}} e^{-\frac{\epsilon_{s}}{\tau}}

Directly from Z - the Partition Function, we can define the Free Energy -F.

F = − τlnZ

The Free Energy represent the struggle between two opposite tendencies of any system, the rise in the entropy (from the second law) and the decline of the energy (from mechanics). In previous course we see how from this definition we can obtain the other form of F:

F = U − στ

which represent the struggle of these two tendencies. Why we introduced new 'additional' quantity? The reason is that this new function is minimal for a system in contact with heat reservoir.

The differential form of F is:

dF = dU − d(τσ)

Previously we found dU = τdσ − PdV, therefore: dF = − σdτ − pdV

[edit] Grand Canonical Ensemble

In this ensemble we attach our system to a heat-particles reservoir in order to fix the temperature and the chemical potential of our system. The chemical potential is defined as \mu=\left(\frac{dF}{dN}\right)_{V,\tau}. In previous ensembles we fixed number of particles N in the system. Now we let it change, so now we should rectify the expression of the energy and all its derivatives.

\begin{matrix} &dU=\tau d\sigma-pdV+\mu dN \\ &dF=-\sigma d\tau-pdV+\mu dN \\ &d\sigma=\frac{dU}{\tau}+\frac{p}{\tau}dV -\frac{\mu}{\tau}dN \end{matrix}

We start by defining the probability of a certain state as:

P(N,\left\{s_{i}\right\})=\frac{1}{\zeta}e^{\frac{\mu N-\epsilon_{s}}{\tau}}

where

\zeta=\sum_{N}\sum_{\left\{s\right\}}e^{\frac{\mu N-\epsilon_{s}}{\tau}}

Directly from ζ, the Grand Partition Function, we shall derive the definition of the Grand Thermodynamical Potential, Ω.

Ω = − τlnζ

Another form of Ω is: Ω = F − μN. The differential form of Ω is:

Ω = − σdτ − pdV − Ndμ

There are other possible ensembles, some of them we shall see in this course.

[edit] Gibbs Potential

We should now elaborate a specific characteristic of physical parameters of a thermodynamic system. An extensive parameter is a parameter whose value is proportional to the size of the system (number of particles), an intensive parameter is a parameter that is independent of the system size.

Let's summarize the thermodynamic potential we have seen:

  • U - Energy
  • F - Free Energy
  • H - Enthalpy
  • Ω - Grand Potential

We have expressed all of these potentials in a differential form, where the number of elements depends on the number of ways to change the system's energy. Each term involves one extensive parameter and one intensive parameter. The pairs we have encountered so far are (τ,σ), (p,V), (μ,N), (B,M), where the first element of each pair is the intensive parameter and the second is the extensive one. The parameters under differential sign have a special meaning: when they are fixed, this thermo potential tends to minimum/maximum. For example:

d\sigma=\frac{dU}{\tau}+\frac{p}{\tau}dV -\frac{\mu}{\tau}dN \Rightarrow \sigma\uparrow max, when U,V,N =

const.

or

dF=-\sigma d\tau-pdV+\mu dN \Rightarrow F\downarrow min when τ,V,N = const.

In order to get from one potential to another one we simply need to add the proper pair of parameters to the first potential. For example, let's construct an ensemble where we fix the temperature, the number of particles and the pressure of the system. Like in the previous ensembles we'll look for a matching potential for our new ensemble and we shall see that our matching potential tends to a minimum.

\begin{matrix} &G(\tau,p,N)=U(V,\sigma,N)+pV-\tau\sigma=H(p,\sigma,N)-\tau\sigma=F(V,\tau,N)+pV \\ &dG=-\sigma d\tau-Vdp+\mu dN \end{matrix}

We attach to the system a heat-particle reservoir and will show that Gibbs potential is minimum at τ, p and N - constant.

The total entropy σR + S increase since R + S is a closed system.

We use Taylor expansion:

\begin{matrix} &\sigma_{R}(U_{0}-U,V_{0}-V)+\sigma_{R}(U,V)\ \uparrow max \\ &\sigma_{R}(U_{0},V_{0})-\frac{\partial \sigma_{R}}{\partial U}|_{U_{0}}\ U-\frac{\partial \sigma_{R}}{\partial V}|_{V_{0}}\ V+\sigma_{S}\ \uparrow max \end{matrix}

Using Maxwell relations we get:

\begin{matrix} &\left(\frac{\partial\sigma_{R}}{\partial U}\right)_{N,V}=\frac{1}{\tau}\\ &\left(\frac{\partial\sigma_{R}}{\partial U}\right)_{N,V}=\frac{p}{\tau} \\ &\underbrace{\sigma_{R}(U_{0},V_{0})}_{constant}-\frac{U}{\tau}-p\frac{V}{\tau}+\sigma_{S}\ \uparrow max \end{matrix}

Now we multiply by − τ, so the expression should tend to minimum:

G=U+pV-\tau\sigma\ \downarrow min

We call this potential Gibbs Free Energy.

[edit] Intensive Potentials

All of our potentials so far are extensive quantities. Sometimes we want to characterize the system by an intensive quantity with a meaning of thermodynamic potential. The easiest way define it is to divide the potential by an extensive parameter. If we divide the potential by, for example, N we'll get the potential per particle, which is an intensive size. We shall see now that in the special case of the potential G we'll get:

\mu=\left(\frac{\partial G}{\partial N} \right)_{p,\tau}=\frac{G}{N}

i.e. μ is Gibbs free energy per particle

We start with the Energy (U):

U=(\sigma,V,N)=NU(\frac{\sigma}{N},\frac{V}{N},1)

The above equation simply states that the energy of a system is equal to the number of particles times the energy of one particle.

The same can be done with the Free Energy (F):

F(\tau,V,N)=NF(\tau,\frac{V}{N},1)=Nf_{1}(\tau,\frac{V}{N})

(The temperature is an intensive quantity so it's the same for every particle) Now we do the same to Gibbs Free Energy:

G(τ,p,N) = NG(τ,p,1) = Ng1(τ,p)

From the differential form of G we calculate the Chemical Potential:

\mu=\left(\frac{\partial G}{\partial N}\right)_{p,\tau}=g_{1}(\tau,p)\ \Rightarrow\ \mu=\frac{G}{N}

From here we receive:

\begin{matrix} &G=F+pV=U+pV-\tau\sigma=\mu N \\ &\Omega=F-\mu N=F-G=-pV \end{matrix}


[edit] Phase transitions:

(example of liquid-vapor transition)

Consider vessel which contain water vapor with volume V and at low pressure so that the gas is an ideal gas. We start decrease the volume so the pressure increases: (p\approx\frac{1}{V}).

In a certain point a small layer of water will appear and from this point, the following decreasing of volume wil not change the pressure, it will be constant. So what happens in the system? If we continue to reduce the volume, the water level will rise, but the pressure will remain constant. When all the vapor turn into water, reducing the volume will increase the pressure drastically and the proportion between the pressure and volume will change. In range of constant pressure we observe coexistence of two phases: liquid and gas.

The process we described above is done at a constant temperature. If we raise the temperature, the range of phase separation will shrink. But, if we raise the temperature high enough, we will have only a point of constant pressure. This temperature is called critical temperature, and for a phase transformation process at this temperature we don't have phase separation, we have everytime coexistence of gas and liquid, this named - fluid.

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