SklogWiki - User contributions [en]

Legendre transform

2007-05-26T15:36:06Z

Cuesta: New page: http://en.wikipedia.org/wiki/Legendre_transform Legendre transform

[[http://en.wikipedia.org/wiki/Legendre_transform Legendre transform]]

Heaviside step distribution

2007-05-26T15:10:29Z

Cuesta:

The '''Heaviside step distribution''' is defined by

:<math>
H(x) = \left\{
\begin{array}{ll}
0 & x < 0 \\
1 & x > 0
\end{array} \right.
</math>
[[category:mathematics]]

Replica method

2007-05-26T15:02:09Z

Cuesta:

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=\lim_{s\to 0}\frac{d}{ds}\left(\frac{Z^{\rm rep}(s)}{Z_0}\right) </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 the <math>s</math> identical non-interacting replicas of the fluid. Since <math>Z_0=Z^{\rm rep}(0)</math>, then

:<math>\lim_{s\to 0}\frac{d}{ds}[-\beta A^{\rm rep}(s)]=\lim_{s\to 0}\frac{d}{ds}\log Z^{\rm rep}(s)=\lim_{s\to 0}\frac{\frac{d}{ds}Z^{\rm rep}(s)}{Z^{\rm rep}(s)}=\lim_{s\to 0}\frac{\frac{d}{ds}Z^{\rm rep}(s)}{Z_0}=-\beta\overline{A}_1.</math>

Thus the relation between the [[Helmholtz energy function]] of the non-equilibrium partially frozen system and the replicated (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)]

Virial coefficients of model systems

2007-02-20T13:26:56Z

Cuesta:

The virial equation of state is used to describe the behavior of diluted gases.
It is usually written as an expansion of the compresiblity factor, <math> Z </math> in terms of either the
density or the pressure. In the first case:

<math> \frac{p V}{N k_B T } = Z = 1 + \sum_{k=2}^{\infty} B_k(T) \rho^{k-1}.
</math>

where

* <math> p </math> is the pressure

*<math> V </math> is the volume

*<math> N </math> is the number of molecules

*<math> \rho \equiv \frac{N}{V} </math> is the (number) density

*<math> B_k </math> is called the k-th virial coefficient

Canonical ensemble

2007-02-20T13:25:09Z

Cuesta: /* Free energy and Partition Function */

Canonical Ensemble:

Variables:

* Number of Particles, <math> N </math>

* Volume, <math> V </math>

* Temperature, <math> T </math>

== Partition Function ==

''Classical'' Partition Function (one-component system) in a three-dimensional space: <math> Q_{NVT} </math>

<math> Q_{NVT} = \frac{V^N}{N! \Lambda^{3N} } \int d (R^*)^{3N} \exp \left[ - \beta U \left( V, (R^*)^{3N} \right) \right] </math>

where:

* <math> \Lambda </math> is the [[de Broglie wavelength]] (depends on the temperature)

* <math> \beta = \frac{1}{k_B T} </math>, with <math> k_B </math> being the [[Boltzmann constant]]

* <math> U </math> is the potential energy, which depends on the coordinates of the particles (and on the interaction model)

* <math> \left( R^*\right)^{3N} </math> represent the 3N position coordinates of the particles (reduced with the system size): i.e. <math> \int d (R^*)^{3N} = 1 </math>

== Free energy and Partition Function ==

The [[Helmholtz energy function|Helmholtz free energy ]]is related to the canonical partition function as:

<math> F\left(N,V,T \right) = - k_B T \log Q_{NVT} </math>