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Heat capacity

2015-11-25T12:00:11Z

129.69.120.39: /* Excess heat capacity */ internal energy has other units than heat capacity. d_v=partial U/partialT

The '''heat capacity''' is defined as the differential of [[heat]] with respect to the [[temperature]] <math>T</math>,

:<math>C := \frac{\delta Q}{\partial T} = T \frac{\partial S}{\partial T}</math>

where <math>Q</math> is [[heat]] and <math>S</math> is the [[entropy]].
==At constant volume==
From the [[first law of thermodynamics]] one has
:<math>\left.\delta Q\right. = dU + pdV</math>
thus at constant volume, denoted by the subscript <math>V</math>, then <math>dV=0</math>,
:<math>C_V := \left.\frac{\delta Q}{\partial T} \right\vert_V = \left. \frac{\partial U}{\partial T} \right\vert_V </math>
==At constant pressure==
At constant [[pressure]] (denoted by the subscript <math>p</math>),
:<math>C_p := \left.\frac{\delta Q}{\partial T} \right\vert_p =\left.\frac{\partial H}{\partial T} \right\vert_p= \left. \frac{\partial U}{\partial T} \right\vert_p + p \left.\frac{\partial V}{\partial T} \right\vert_p</math>

where <math>H</math> is the [[enthalpy]].
The difference between the heat capacity at constant pressure and the heat capacity at constant volume is given by
:<math>C_p -C_V = \left( p + \left. \frac{\partial U}{\partial V} \right\vert_T \right) \left. \frac{\partial V}{\partial T} \right\vert_p</math>
==Adiabatic index==
Sometimes the ratio of heat capacities is known as the ''adiabatic index'':

:<math>\gamma = \frac{C_p}{C_V}</math>

==Excess heat capacity==
In a classical system the excess heat capacity for a monatomic fluid is given by subtracting the [[Ideal gas: Energy |ideal internal energy]] (which is kinetic in nature)

:<math>C_v^{ex} = C_v - \frac{3}{2}Nk_B</math>

in other words the excess heat capacity is associated with the component of the internal energy due to the intermolecular potential, and for that reason it is also known as the ''configurational'' heat capacity. Given that the excess internal energy for a pair potential is given by (Eq. 2.5.20 in <ref>J-P. Hansen and I. R. McDonald "Theory of Simple Liquids", Academic Press (2006) (Third Edition) ISBN 0-12-370535-5 </ref>):

:<math>U^{ex} = 2\pi N \rho \int_0^{\infty} \Phi_{12}(r) g(r) r^2 ~{\rm d}{\mathbf r}</math>

where <math>\Phi_{12}(r)</math> is the [[intermolecular pair potential]] and <math>g(r)</math> is the [[radial distribution function]],
one has

:<math>C_v^{ex} = 2\pi N \rho \int_0^{\infty} \Phi_{12}(r) \left. \frac{\partial g(r)}{\partial T} \right\vert_V r^2 ~{\rm d}{\mathbf r} </math>

For many-body distribution functions things become more complicated <ref>[http://dx.doi.org/10.1063/1.468220 Ben C. Freasier, Adam Czezowski, and Richard J. Bearman "Multibody distribution function contributions to the heat capacity for the truncated Lennard‐Jones fluid", Journal of Chemical Physics '''101''' pp. 7934-7938 (1994)]</ref>.
===Rosenfeld-Tarazona expression===
Rosenfeld and Tarazona
<ref>[http://dx.doi.org/10.1080/00268979809483145 Yaakov Rosenfeld and Pedro Tarazona "Density functional theory and the asymptotic high density expansion of the free energy of classical solids and fluids", Molecular Physics '''95''' pp. 141-150 (1998)]</ref>
<ref>[http://dx.doi.org/10.1063/1.4827865 Trond S. Ingebrigtsen , Arno A. Veldhorst , Thomas B. Schrøder and Jeppe C. Dyre "Communication: The Rosenfeld-Tarazona expression for liquids’ specific heat: A numerical investigation of eighteen systems", Journal of Chemical Physics '''139''' 171101 (2013)]</ref>
used [[fundamental-measure theory]] to obtain a ''unified analytical description'' of classical bulk solids and fluids, one result being:

:<math>C_v^{ex} \propto T^{-2/5}</math>

==Liquids==
The calculation of the heat capacity in liquids is more difficult than in gasses or solids <ref>[http://dx.doi.org/10.1063/1.1667469 Claudio A. Cerdeiriña, Diego González-Salgado, Luis Romani, María del Carmen Delgado, Luis A. Torres and Miguel Costas "Towards an understanding of the heat capacity of liquids. A simple two-state model for molecular association", Journal of Chemical Physics '''120''' pp. 6648-6659 (2004)]</ref>.
Recently an expression for the energy of a liquid has been developed (Eq. 5 of <ref>[http://dx.doi.org/10.1038/srep00421 D. Bolmatov, V. V. Brazhkin and K. Trachenko "The phonon theory of liquid thermodynamics", Scientific Reports '''2''' Article number: 421 (2012)]</ref>):

:<math>E = NT \left( 1 + \frac{\alpha T}{2}\right) \left( 3D \left( \frac{\hbar \omega_D}{T} \right) -\left( \frac{\omega_F}{\omega_D} \right)^3 D\left( \frac{\hbar \omega_F}{T}\right) \right)</math>

where <math>\omega_F</math> is the [[Frenkel frequency]], <math>\omega_D</math> is the [[Debye frequency]], <math>D</math> is the [[Debye function]], and <math>\alpha</math>
is the [[thermal expansion coefficient]]. The differential of this energy with respect to temperature provides the heat capacity.

==Solids==
====Petit and Dulong====
<ref>Alexis-Thérèse Petit and Pierre-Louis Dulong "Recherches sur quelques points importants de la Théorie de la Chaleur", Annales de Chimie et de Physique '''10''' pp. 395-413 (1819)</ref>
====Einstein====
====Debye====
A low temperatures on has

:<math>c_v = \frac{12 \pi^4}{5} n k_B \left( \frac{T}{\Theta_D} \right)^3</math>

where <math>k_B</math> is the [[Boltzmann constant]], <math>T</math> is the [[temperature]] and <math>\Theta_D</math> is an empirical parameter known as the Debye temperature.

==See also==
*[[Ideal gas: Heat capacity | Heat capacity of an ideal gas]]
*[[Yang-Yang anomaly]]

==References==
<references/>
[[category: classical thermodynamics]]

Wang-Landau method

2010-11-17T14:30:02Z

129.69.120.39:

The '''Wang-Landau method''' was proposed by F. Wang and D. P. Landau (Ref. 1-2) to compute the density of
states, <math> \Omega (E) </math>, of [[Potts model|Potts models]];
where <math> \Omega(E) </math> is the number of [[microstate |microstates]] of the system having energy
<math> E </math>.

== Sketches of the method ==
The Wang-Landau method, in its original version, is a [[Computer simulation techniques |simulation technique]] designed to achieve a uniform sampling of the energies of the system in a given range.
In a standard [[Metropolis Monte Carlo|Metropolis Monte Carlo]] in the [[canonical ensemble|canonical ensemble]]
the probability of a given [[microstate]], <math> X </math>, is given by:

:<math> P(X) \propto \exp \left[ - E(X)/k_B T \right] </math>;

whereas for the Wang-Landau procedure one can write:

:<math> P(X) \propto \exp \left[ f(E(X)) \right] </math> ;

where <math> f(E) </math> is a function of the energy. <math> f(E) </math> changes
during the simulation in order produce a predefined distribution of energies (usually
a uniform distribution); this is done by modifying the values of <math> f(E) </math>
to reduce the probability of the energies that have been already ''visited'', i.e.
If the current configuration has energy <math> E_i </math>, <math> f(E_i) </math>
is updated as:

:<math> f^{new}(E_i) = f(E_i) - \Delta f </math> ;

where it has been considered that the system has discrete values of the energy (as happens in [[Potts model|Potts Models]]), and <math> \Delta f > 0 </math>.

Such a simple scheme is continued until the shape of the energy distribution
approaches the one predefined. Notice that this simulation scheme does not produce
an equilibrium procedure, since it does not fulfill [[detailed balance]]. To overcome
this problem, the Wang-Landau procedure consists in the repetition of the scheme
sketched above along several stages. In each subsequent stage the perturbation
parameter <math> \Delta f </math> is reduced. So, for the last stages the function <math> f(E) </math> hardly changes and the simulation results of these last stages can be considered as a good description of the actual equilibrium system, therefore:

:<math> g(E) \propto e^{f(E)} \int d X_i \delta( E, E_i ) = e^{f(E)} \Omega(E)</math>;

where <math> E_i = E(X_i) </math>, <math> \delta(x,y) </math> is the
[[Kronecker delta|Kronecker Delta]], and <math> g(E) </math> is the fraction of
microstates with energy <math> E </math> obtained in the sampling.

If the probability distribution of energies, <math> g(E) </math>, is nearly flat (if a uniform distribution of energies is the target), i.e.
: <math> g(E_i) \simeq 1/n_{E} ; </math>; for each value <math> E_i </math> in the selected range,
with <math> n_{E} </math> being the total number of discrete values of the energy in the range, then the density of
states will be given by:

:<math> \Omega(E) \propto \exp \left[ - f(E) \right] </math>

=== Microcanonical thermodynamics ===

Once one knows <math> \Omega(E) </math> with accuracy, one can derive the thermodynamics
of the system, since the [[entropy|entropy]] in the [[microcanonical ensemble|microcanonical ensemble]] is given by:

:<math> S \left( E \right) = k_{B} \log \Omega(E) </math>

where <math> k_{B} </math> is the [[Boltzmann constant | Boltzmann constant]].

== Extensions ==
The Wang-Landau method has inspired a number of simulation algorithms that
use the same strategy in different contexts. For example:
* [[Inverse Monte Carlo|Inverse Monte Carlo]] methods (Refs 4-6)
* [[Computation of phase equilibria]] of fluids (Refs 7-9)
* Control of polydispersity by chemical potential ''tuning'' (Ref 6)

=== Phase equilibria ===

In the original version one computes the [[entropy|entropy]] of the system as a function of
the [[internal energy|internal energy]], <math> E </math>, for fixed conditions of volume,
and number of particles.
In Refs (7-9) it is shown how the procedure can be applied to compute other thermodynamic
potentials that can be used later to locate [[phase transitions]]. For instance one
can compute the [[Helmholtz energy function | Helmholtz energy function ]],
<math> A \left( N | V, T \right) </math> as a function of the number of particle <math> N </math>
for fixed conditions of volume, <math> V </math>, and [[temperature|temperature]], <math> T </math>.

=== Refinement of the results ===
It can be convenient (Refs 7-8) to supplement the Wang-Landau algorithm, which does not fulfil detailed balance,
with an equilibrium simulation. In this equilibrium simulation one can use
the final result for <math> f\left( E \right) </math> (or <math> f\left( N \right) </math>) extracted from
the Wang-Landau technique as a fixed function to weight
the probability of the different configurations.
Such a strategy simplifies the estimation of error bars, provides a good test of the results consistency,
and can be used to refine the numerical results.
==See also==
*[[Statistical-temperature simulation algorithm]]
==References==
#[http://dx.doi.org/10.1103/PhysRevLett.86.2050 Fugao Wang and D. P. Landau "Efficient, Multiple-Range Random Walk Algorithm to Calculate the Density of States", Physical Review Letters '''86''' pp. 2050-2053 (2001)]
#[http://dx.doi.org/10.1103/PhysRevE.64.056101 Fugao Wang and D. P. Landau "Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram", Physical Review E '''64''' 056101 (2001)]
#[http://dx.doi.org/10.1119/1.1707017 D. P. Landau, Shan-Ho Tsai, and M. Exler "A new approach to Monte Carlo simulations in statistical physics: Wang-Landau sampling", American Journal of Physics '''72''' pp. 1294-1302 (2004)]
#[http://dx.doi.org/10.1103/PhysRevE.68.011202 N. G. Almarza and E. Lomba, "Determination of the interaction potential from the pair distribution function: An inverse Monte Carlo technique", Physical Review E '''68''' 011202 (6 pages) (2003)]
#[http://dx.doi.org/10.1103/PhysRevE.70.021203 N. G. Almarza, E. Lomba, and D. Molina. "Determination of effective pair interactions from the structure factor", Physical Review E '''70''' 021203 (5 pages) (2004)]
#[http://dx.doi.org/10.1063/1.1626635 Nigel B. Wilding "A nonequilibrium Monte Carlo approach to potential refinement in inverse problems", Journal of Chemical Physics '''119''', 12163 (2003) ]
#[http://dx.doi.org/10.1103/PhysRevE.71.046132 E. Lomba, C. Martín, and N. G. Almarza, "Simulation study of the phase behavior of a planar Maier-Saupe nematogenic liquid", Physical Review E E '''71''' 046132 (2005) ]
#[http://dx.doi.org/10.1063/1.2748043 E. Lomba, N. G. Almarza, C. Martín, and C. McBride, "Phase behavior of attractive and repulsive ramp fluids: Integral equation and computer simulation studies", Journal of Chemical Physics '''126''' 244510 (2007) ]
#[http://dx.doi.org/10.1063/1.2794042 Georg Ganzenmüller and Philip J. Camp "Applications of Wang-Landau sampling to determine phase equilibria in complex fluids", Journal of Chemical Physics '''127''' 154504 (2007)]
#[http://dx.doi.org/10.1063/1.2803061 R. E. Belardinelli and V. D. Pereyra "Wang-Landau algorithm: A theoretical analysis of the saturation of the error", Journal of Chemical Physics '''127''' 184105 (2007)]
#[http://dx.doi.org/10.1103/PhysRevE.75.046701 R. E. Belardinelli and V. D. Pereyra "Fast algorithm to calculate density of states", Physical Review E '''75''' 046701 (2007)]
[[category: Monte Carlo]]
[[category: computer simulation techniques]]