Introduction to Mathematical Physics/Statistical physics/Constraint relaxing

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We have defined at section secmaxient external variables, fixed by the exterior, and internal variables free to fluctuate around a fixed mean. Consider a system L being described by N+N' internal variables \index{constraint} n_1,\dots,n_N,X_1,\dots,X_{N'}. This system has a partition function Z^{L}. Consider now a system F, such that variables n_iare this time considered as external variables having value N_i. This system F has (another) partition function we call Z^{F}. System L is obtained from system F by constraint relaxing. Here is theorem that binds internal variables n_i of system L to partition function Z^F of system F :

Theorem:

Values n_i the most probable in system L where n_i are free to fluctuate are the values that make zero the differential of partition function Z^{F} where the n_i's are fixed.

Proof:

Consider the description where the n_i's are free to fluctuate. Probability for event n_1=N_1,\cdots,n_N=N_N occurs is:


P(n_1=N_1,\cdots,n_N=N_N)=\sum_{(l)/n_1=N_1,\cdots,n_N=N_N}P^{L}_l

So

\begin{matrix}
P(n_1=N_1,\cdots,n_N=N_N) = \\
&&\frac{1}{Z^{nl}} e^
{-\lambda_{n_1}N_1-\cdots-\lambda_{n_N}N_N} \sum_{(l)/n_1 =
N_1,\cdots,n_N=N_N} e^ {-\lambda_{X_1}\bar X_1-\lambda_{X_2}\bar X_2} 
\end{matrix}

Values the most probable make zero differential dP(n_1=N_1,\cdots,n_N=N_N) (this corresponds to the maximum of a (differentiable) function P.). So


d\log(Z^{F})=\sum_i \frac{\partial \log(Z^{F})}{\partial n_i}dn_i=0

Remark:

This relation is used in chemistry: it is the fundamental relation of chemical reaction. In this case, the N variables n_i represent the numbers of particles of the N species i and N'=2 with X_1=E (the energy of the system) and X_2=V (the volume of the system). Chemical reaction equation gives a binding on variables n_i that involves stoechiometric coefficients.

Let us write a Gibbs-Duheim type relation \index{Gibbs-Duheim relation}:


S^{F}/k=\lambda_{X_1}\bar X_1+\lambda_{X_2}\bar X_2+\ln (Z^{F})


S^{L}/k=\lambda_{X_1}\bar X_1+\lambda_{X_2}\bar X_2+\lambda_{n_1}\bar{n_1}+\cdots+\ln (Z^{L})

At thermodynamical equilibrium S^{F}=S^{L}, so:


\lambda_{n_i}=\frac{\partial \ln (Z^{F})}{\partial n_i}

Example:

This last equality provides a way to calculate the chemical potential of the system.\index{chemical potential}


\mu_i=\frac{\partial \ln (Z^{F})}{\partial n_i}

In general one notes -\ln(Z^{F})=G.

Example:

Consider the case where variables n_i are the numbers of particles of species i. If the particles are independent, energy associated to a state describing the N particles (the set of particles of type i being in state l_i) is the sum of the N energies associated to states l_i. Thus:


\ln Z^F(\lambda_{X_1},\lambda_{X_2},n_1,n_2)=\ln
Z^F_1(\lambda_{X_1},\lambda_{X_2},n_1)+\dots+\ln
Z^F_N(\lambda_{X_1},\lambda_{X_2},n_N)

where Z^F_i(\lambda_{X_1},\lambda_{X_2},n_i) represents the partition function of the system constituted only by particles of type i, for which the value of variable n_i is fixed. So:


\frac{\partial \ln Z^F_1}{\partial n_1} dn_1+\dots+\frac{\partial \ln
Z^F_N}{\partial n_N} dn_N=0

Remark:

Setting \lambda_1=\beta,\lambda_2=\beta p and \lambda_{n_1}=-\beta\mu with G=-k_BT\ln Z^{nf}, we have G(T,p,n)=E+pV-TS and G'(T,p,\mu)=E+pV-TS-\mu-{n_1}. This is a Gibbs-Duheim relation.

Example:

We propose here to prove the Nernst formula\index{Nernst formula} describing an oxydo-reduction reaction.\index{oxydo-reduction} This type of chemical reaction can be tackled using previous formalism. Let us precise notations in a particular case. Nernst formula demonstration that we present here is different form those classically presented in chemistry books. Electrons undergo a potential energy variation going from solution potential to metal potential. This energy variation can be seen as the work got by the system or as the internal energy variation of the system, depending on the considered system is the set of the electrons or the set of the electrons as well as the solution and the metal. The chosen system is here the second. Consider the free enthalpy function  G(T,p,n_i,E_p). Variables n_i and E_p are free to fluctuate. They have values such that G is minimum. let us calculate the differential of G:

dG(T,p,n_i,E_p)=\frac{\partial G}{\partial T} dT + \frac{\partial G}{\partial p} dp + \sum_i \frac{\partial G}{\partial n_i} dn_i + \frac{\partial G}{\partial E_p} dE_p

Using definition\footnote{the internal energy U is the sum of the kinetic energy and the potential energy, so as G can be written itself as a sum:

G=U+pV-TS

}

of G:

\frac{\partial G}{\partial E_p}=1

one gets:

0=\sum_i\mu_idn_i+dE_p

If we consider reaction equation:


Ox+n\bar e\longrightarrow Red

0=\sum_i \mu_i \nu_i d\xi +nq(V_{red}-V_{ox})

So:

V_{red}-V_{ox}=\frac{\sum_i\nu_i\mu_i}{nF}

dG can only decrease. Spontaneous movement of electrons is done in the sense that implies dE_p < 0. As dE_p^{ext}=-dE_p^{int} we chose as definition of electrical potential:

V^{ext}=-V^{int}

Nernst formula deals with the electrical potential seen by the exterior.