Replica method

2007-05-24T11:05:27Z

80.38.96.35:

The [[Helmholtz energy function]] of fluid in a matrix of configuration
<math>\{ q^{N_0} \}</math> in the Canonical (<math>NVT</math>) ensemble is given by:

:<math>- \beta A_1 (q^{N_0}) = \log Z_1 (q^{N_0})
= \log \left( \frac{1}{N_1!}
\int \exp [- \beta (H_{11}(r^{N_1}) + H_{10}(r^{N_1}, q^{N_0}) )]~d \{ r \}^{N_1} \right)</math>

where <math>Z_1 (q^{N_0})</math> is the fluid [[partition function]], and <math>H_{11}</math>, <math>H_{10}</math> and <math>H_{00}</math>
are the pieces of the Hamiltonian corresponding to the fluid-fluid, fluid-matrix and matrix-matrix interactions. Assuming that the matrix is a configuration of a given fluid, with interaction hamiltonian <math>H_{00}</math>, we can average over matrix configurations to obtain

:<math>- \beta \overline{A}_1 = \frac{1}{N_0!Z_0} \int \exp [-\beta_0 H_{00} ( q^{N_0})] ~ \log Z_1 (q^{N_0}) ~d \{ q \}^{N_0}</math>

(see Refs. 1 and 2)

:An important mathematical trick to get rid of the logarithm inside of the integral is to use the mathematical identity

:<math>\log x = \lim_{s \rightarrow 0} \frac{{\rm d}}{{\rm d}s}x^s</math>.

One can apply this trick to the <math>\log Z_1</math> we want to average, and replace the resulting power <math>(Z_1)^s</math> by <math>s</math> copies of the expression for <math>Z_1</math> ''(replicas)''. The result is equivalent to evaluate <math>\overline{A}_1</math> as

:<math> -\beta\overline{A}_1=\frac{Z^{\rm rep}(s)}{Z_0} </math>,

where <math>Z^{\rm rep}(s)</math> is the partition function of a mixture with Hamiltonian

:<math>\beta H^{\rm rep} (r^{N_1}, q^{N_0})
= \frac{\beta_0}{\beta}H_{00} (q^{N_0}) + \sum_{\lambda=1}^s
\left( H_{01}^\lambda (r^{N_1}_\lambda, q^{N_0}) + H_{11}^\lambda (r^{N_1}_\lambda, q^{N_0})\right).</math>

This Hamiltonian describes a completely equilibrated system of <math>s+1</math> components; the matrix and <math>s</math> identical non-interacting copies (''replicas'') of the fluid. Thus the relation between the [[Helmholtz energy function]] of the non-equilibrium partially frozen
and the replica (equilibrium) system is given by

:<math>- \beta \overline{A}_1 = \lim_{s \rightarrow 0} \frac{{\rm d}}{{\rm d}s} [- \beta A^{\rm rep} (s) ]
</math>.
==References==
#[http://dx.doi.org/10.1088/0305-4608/5/5/017 S F Edwards and P W Anderson "Theory of spin glasses",Journal of Physics F: Metal Physics '''5''' pp. 965-974 (1975)]
#[http://dx.doi.org/10.1088/0305-4470/9/10/011 S F Edwards and R C Jones "The eigenvalue spectrum of a large symmetric random matrix", Journal of Physics A: Mathematical and General '''9''' pp. 1595-1603 (1976)]

Partition function

2007-05-24T10:41:33Z

80.38.96.35:

The '''partition function''' of a system in contact with a thermal bath
at temperature <math>T</math> is the normalization constant of the [[Boltzmann distribution]]
function, and therefore its expression is given by

:<math>Z(T)=\int \Omega(E)\exp(-E/k_BT)\,dE</math>,

where <math>\Omega(E)</math> is the [[density of states]] with energy <math>E</math> and <math>k_B</math>
the [[Boltzmann constant]].

The partition function of a system is related to the [[Helmholtz energy function]] through the formula

:<math>\left.A\right.=-k_BT\log Z.</math>

This connection can be derived from the fact that <math>k_B\log\Omega(E)</math> is the
[[entropy]] of a system with total energy <math>E</math>. This is an [[extensive magnitude]] in the
sense that, for large systems (i.e. in the [[thermodynamic limit]], when the number of
particles <math>N\to\infty</math>
or the volume <math>V\to\infty</math>), it is proportional to <math>N</math> or <math>V</math>.
In other words, if we assume <math>N</math> large, then

:<math>\left.k_B\right. \log\Omega(E)=Ns(e),</math>

where <math>s(e)</math> is the entropy per particle in the [[thermodynamic limit]], which is
a function of the energy per particle <math>e=E/N</math>. We can
therefore write

:<math>\left.Z(T)\right.=N\int \exp\{N(s(e)-e/T)/k_B\}\,de.</math>

Since <math>N</math> is large, this integral can be performed through [[steepest descent]],
and we obtain

:<math>\left.Z(T)\right.=N\exp\{N(s(e_0)-e_0/k_BT)\}</math>,

where <math>e_0</math> is the value that maximizes the argument in the exponential; in other
words, the solution to

:<math>\left.s'(e_0)\right.=1/T.</math>

This is the thermodynamic formula for the inverse temperature provided <math>e_0</math> is
the mean energy per particle of the system. On the other hand, the argument in the exponential
is

:<math>\frac{1}{k_BT}(TS(E_0)-E_0)=-\frac{A}{k_BT}</math>

the thermodynamic definition of the [[Helmholtz energy function]]. Thus, when <math>N</math> is large,

:<math>\left.A\right.=-k_BT\log Z.</math>

The internal energy is given by

:<math>U=k_B T^{2} \frac{\partial \log Z(T)}{\partial T}</math>

These equations provides a link between [[Classical thermodynamics | classical thermodynamics]] and
[[Statistical mechanics | statistical mechanics]]

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Replica method

Partition function