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Virial (disambiguation)

2010-10-07T09:42:50Z

211.121.168.82: English corrections

The '''virial''' of a central [[ Intermolecular_pair_potential | interaction potential]] <math>v(r)</math> is defined as

:<math>V(r)= \frac{dv(r)}{dr} r = - f(r) r .</math>

The virial features prominently in [[classical thermodynamics]] (for example, see [[Virial_equation_of_state | virial coefficients]]) and in simulation (see: [[virial pressure]]).
==See also==
*[[Virial equation of state]]
*[[Virial pressure]]
*[[Virial theorem]]
[[category: classical mechanics]]

Virial (disambiguation)

2010-10-07T09:41:48Z

211.121.168.82: English corrections

The '''virial''' of a central [[ Intermolecular_pair_potential | interaction potential]] <math>v(r)</math> is defined by

:<math>V(r)= \frac{dv(r)}{dr} r = - f(r) r .</math>

The virial features prominently in [[classical thermodynamics]] (for example, see [[Virial_equation_of_state | virial coefficients]]) and in simulation (see: [[virial pressure]]).
==See also==
*[[Virial equation of state]]
*[[Virial pressure]]
*[[Virial theorem]]
[[category: classical mechanics]]

Virial pressure

2010-10-07T08:57:58Z

211.121.168.82: Punctuation correction

The '''virial pressure''' is commonly used to obtain the [[pressure]] from a general simulation. It is particularly well suited to [[molecular dynamics]], since [[Newtons laws#Newton's second law of motion |forces]] are evaluated and readily available. For pair interactions, one has:

:<math> p = \frac{ k_B T N}{V} - \frac{ 1 }{ d V } \overline{ \sum_{i<j} {\mathbf f}_{ij} {\mathbf r}_{ij} }, </math>

where <math>p</math> is the pressure, <math>T</math> is the [[temperature]], <math>V</math> is the volume and <math>k_B</math>is the [[Boltzmann constant]].
In this equation one can recognize an [[Equation of State: Ideal Gas |ideal gas]] contribution, and a second term due to the [[virial]]. The overline is an average, which would be a time average in molecular dynamics, or an ensemble average in [[Monte Carlo]]; <math>d</math> is the dimension of the system (3 in the "real" world). <math> {\mathbf f}_{ij} </math> is the force '''on''' particle <math>i</math> exerted '''by''' particle <math>j</math>, and <math>{\mathbf r}_{ij}</math> is the vector going '''from''' <math>i</math> '''to''' <math>j</math>: <math>{\mathbf r}_{ij} = {\mathbf r}_j - {\mathbf r}_i</math>.

This relationship is readily obtained by writing the [[partition function]] in "reduced coordinates", i.e. <math>x^*=x/L</math>, etc, then considering a "blow-up" of the system by changing the value of <math>L</math>. This would apply to a simple cubic system, but the same ideas can also be applied to obtain expressions for the [[stress | stress tensor]] and the [[surface tension]], and are also used in [[constant-pressure Monte Carlo]].

If the interaction is central, the force is given by
:<math> {\mathbf f}_{ij} = - \frac{{\mathbf r}_{ij}}{ r_{ij}} f(r_{ij}) , </math>
where <math>f(r)</math> the force corresponding to the [[Intermolecular pair potential |intermolecular potential]] <math>\Phi(r)</math>:

:<math>-\partial \Phi(r)/\partial r.</math>

For example, for the [[Lennard-Jones model | Lennard-Jones potential]], <math>f(r)=24\epsilon(2(\sigma/r)^{12}- (\sigma/r)^6 )/r</math>. Hence, the expression reduces to
:<math> p = \frac{ k_B T N}{V} + \frac{ 1 }{ d V } \overline{ \sum_{i<j} f(r_{ij}) r_{ij} }. </math>

Notice that most [[Realistic models |realistic potentials]] are attractive at long ranges; hence the first correction to the ideal pressure will be a negative contribution: the [[second virial coefficient]]. On the other hand, contributions from purely repulsive potentials, such as [[hard sphere model | hard spheres]], are always positive.
==See also==
*[[Test volume method]]
[[category: statistical mechanics]]

Gibbs ensemble Monte Carlo

2010-09-22T08:44:26Z

211.121.168.82: English improvements

Phase separation is one of the topics to which [[Computer simulation techniques |simulation techniques]] are increasingly applied. Different procedures are available for this purpose. For the particular case of chain systems, one can employ simulations in the [[Semi-grand ensembles | semi-grand canonical ensemble]], [[histogram reweighting]], or characterization of the [[spinodal curve]] from the study of computed [[collective scattering function]].
The Gibbs ensemble Monte Carlo method has been specifically designed to characterize [[phase transitions]]. It was mainly developed by Panagiotopoulos (Refs. 1 and 2) to avoid the problem of finite size interfacial effects. In this method, an [[Canonical ensemble |NVT]] (or [[Isothermal-isobaric ensemble |NpT]]) ensemble containing two (or more) species is divided into two (or more) boxes. In addition to the usual particle moves in each one of the boxes, the algorithm includes moves steps to change the volume and composition of the boxes at mechanical and chemical equilibrium. Transferring a chain molecule from a box to the other requires the use of an efficient method to insert chains. The [[Configurational bias Monte Carlo | configurational bias method]] is specially recommended for this purpose.
==See also==
*[[Gibbs ensemble]]
==References==
#[http://dx.doi.org/10.1080/00268978700101491 Athanassios Panagiotopoulos "Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble", Molecular Physics '''61''' pp. 813-826 (1987)]
#[http://dx.doi.org/10.1080/00268978800100361 A. Z. Panagiotopoulos, N. Quirke, M. Stapleton and D. J. Tildesley "Phase equilibria by simulation in the Gibbs ensemble: Alternative derivation, generalization and application to mixture and membrane equilibria", Molecular Physics '''61''' pp. 527-545 (1988)]
==External links==
*[http://kea.princeton.edu/jerring/gibbs/ Gibbs ensemble Monte Carlo code] on the [http://www.princeton.edu/che/people/faculty/panagiotopoulos/group/ Panagiotopoulos Group Homepage]
[[category: Monte Carlo]]
[[category: Computer simulation techniques]]