2011-08-23T13:37:59Z

Dduque: Added Cole

'''Equations of state''' are generally expressions that relate the macroscopic observables, or ''state variables'', such as [[pressure]], <math>p</math>, volume, <math>V</math>, and [[temperature]], <math>T</math>.
==General==
*[[Common bulk modulus point]]
*[[Law of corresponding states]]
*[[Linear isothermal regularity]]
*[[Maxwell's equal area construction]]
*[[Tait-Murnaghan relation]]
*[[Zeno line]]

==Virial equations of state==
*[[Virial equation of state]]
*[[Second virial coefficient]]
*[[Virial coefficients of model systems]]
==Semi-empirical equations of state==
Naturally there is the [[Equation of State: Ideal Gas|ideal gas equation of state]]. However, one of the first steps towards a description of realistic substances was the famous [[van der Waals equation of state]]. Since then a plethora of semi-empirical equations have been developed, often in a similar vein to the van der Waals equation of state, each trying to better reproduce the foibles of the many
gasses and/or liquids that are often of industrial interest.
{{columns-list|3|
*[[Amagat equation of state | Amagat]]
*[[Antoine equation of state | Antoine]]
*[[Battelli equation of state |Battelli]]
*[[Beattie-Bridgeman equation of state | Beattie-Bridgeman]]
*[[Benedict, Webb and Rubin equation of state |Benedict, Webb and Rubin ]]
*[[Berthelot equation of state | Berthelot]]
*[[Boltzmann equation of state | Boltzmann]]
*[[Boynton and Bramley equation of state | Boynton and Bramley]]
*[[Brillouin equation of state | Brillouin]]
*[[Clausius equation of state | Clausius]]
*[[Cole equation of state | Cole]]
*[[Dieterici equation of state | Dieterici]]
*[[Dupré equation of state |Dupré]]
*[[Elliott, Suresh, and Donohue equation of state |Elliott, Suresh, and Donohue]]
*[[Fouché equation of state | Fouché]]
*[[Goebel equation of state | Goebel]]
*[[Hirn equation of state |Hirn]]
*[[Jäger equation of state | Jäger]]
*[[Kam equation of state | Kam]]
*[[Lagrange equation of state | Lagrange]]
*[[Leduc equation of state | Leduc]]
*[[Linear isothermal regularity]]
*[[Lorentz equation of state |Lorenz]]
*[[Mie equation of state | Mie]]
*[[Murnaghan equation of state | Murnaghan]]
*[[Natanson equation of state | Natanson]]
*[[Onnes equation of state | Onnes]]
*[[Peczalski equation of state |Peczalski]]
*[[Peng and Robinson equation of state |Peng and Robinson]]
*[[Planck equation of state |Planck]]
*[[Porter equation of state | Porter]]
*[[Rankine equation of state |Rankine]]
*[[Recknagel equation of state | Recknagel]]
*[[Redlich-Kwong equation of state | Redlich-Kwong]]
*[[Reinganum equation of state | Reinganum]]
*[[Sarrau equation of state | Sarrau]]
*[[Schiller equation of state | Schiller]]
*[[Schrieber equation of state | Schrieber]]
*[[Smoluchowski equation of state | Smoluchowski]]
*[[Starkweather equation of state |Starkweather ]]
*[[Tait equation of state | Tait]]
*[[Thiesen equation of state |Thiesen]]
*[[Tumlirz equation of state | Tumlirz]]
*[[van der Waals equation of state | van der Waals]]
*[[Walter equation of state | Walter]]
*[[Wohl equation of state | Wohl]]
*[[Phase diagrams of water | Water equation of state]]
}}

==Other methods==
*[[ASOG (Analytical Solution of Groups)]]
*[[UNIFAC (Universal Functional Activity Coefficient)]]
==Model systems==
Equations of state for [[idealised models]]:
*[[Equation of State: three-dimensional hard dumbbells | Three-dimensional hard dumbbells]]
*[[Equations of state for hard convex bodies| Hard convex bodies]]
*[[Equations of state for hard rods | Hard rods]]
*[[Equations of state for the Gaussian overlap model | Gaussian overlap model]]
*[[Equations of state for the square shoulder model | Square shoulder model]]
*[[Equations of state for the square well model | Square well model]]
*[[Equations of state for the triangular well model | Triangular well model]]
*[[Equations of state for hard spheres]]
*[[Equations of state for crystals of hard spheres]]
*[[Equations of state for hard sphere mixtures]]
*[[Equations of state for hard disks]]
*[[Hard ellipsoid equation of state]]
*[[Lennard-Jones equation of state]]
*[[Fused hard sphere chains#Equation of state | Fused hard sphere chains]]
*[[Tetrahedral hard sphere model#Equation of state|Tetrahedral hard sphere model]]
==Interesting reading==
*[http://dx.doi.org/10.1088/0034-4885/7/1/312 James A. Beattie and Walter H. Stockmayer "Equations of state", Reports on Progress in Physics '''7''' pp. 195-229 (1940)]
*[http://dx.doi.org/10.1021/ie50663a005 K. K. Shah and G. Thodos "A Comparison of Equations of State", Industrial & Engineering Chemistry '''57''' pp. 30-37 (1965)]
*[http://dx.doi.org/10.1088/0034-4885/28/1/306 J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics '''28''' pp. 169-199 (1965)]
'''Books'''
*"Equations of State for Fluids and Fluid Mixtures", Eds. J. V. Sengers, R. F. Kayser, C. J. Peters, and H. J. White Jr., Elsevier (2000) ISBN 0-444-50384-6
[[Category: Results]]

Lees-Edwards boundary conditions

2011-03-10T20:40:03Z

Dduque: cat. added

These conditions are an adaption of standard [[periodic boundary conditions]] to simulations of shear flow (Lees and Edwards, 1972). They are convenient for e.g. simulations of [[Couette flow]].

==References==
<references/>
'''Related reading'''
*[http://epress.anu.edu.au/sm/html/ch06s03.html Couette Flow and Shear Viscosity], in [http://epress.anu.edu.au/sm_citation.html Denis J. Evans and Gary P. Morriss "Statistical Mechanics of Nonequilibrium Liquids" ANU E Press (2007)]
[[category: Computer simulation techniques]]

2010-05-17T15:25:48Z

Dduque: New section

Navier-Stokes equations

2010-05-13T09:54:14Z

Dduque: New page with NS equations

Hydrodynamics

2010-05-13T09:42:52Z

Dduque: New page: '''Hydrodynamics''' is the science of the macroscopic motion of fluid bodies. It is a field theory, relating scalar fields (such as the density and the pressure) with vector fields (such a...

'''Hydrodynamics''' is the science of the macroscopic motion of fluid bodies. It is a field theory, relating scalar fields (such as the density and the pressure) with vector fields (such as the velocity field) and tensor fields (stress tensor field).

== See also ==

[[Navier-Stokes equations]]

Capillary waves

2010-05-13T09:40:59Z

Dduque: Link to (new) hydrodynamics

==Thermal capillary waves==
Thermal '''capillary waves''' are oscillations of an [[interface]] which are thermal in origin. These take place at the molecular level, where only the contribution due to [[surface tension]] is relevant.
Capillary wave theory is a classic account of how thermal fluctuations distort an interface (Ref. 1). It starts from some [[intrinsic surface]] that is distorted. In the Monge representation, the surface is given as <math>z=h(x,y)</math>. An increase in area of the surface causes a proportional increase of energy:

:<math>
E_\mathrm{st}= \sigma \iint dx\, dy\ \sqrt{1+\left( \frac{dh}{dx} \right)^2+\left( \frac{dh}{dy} \right)^2} -1
</math>

for small values of the derivatives (surfaces not too rough):

:<math>
E_\mathrm{st} \approx \frac{\sigma}{2} \iint dx\, dy\ \left[ \left( \frac{dh}{dx} \right)^2+\left( \frac{dh}{dy} \right)^2 \right].
</math>

A [[Fourier analysis]] treatment begins by writing the intrinsic surface as an infinite sum of normal modes:

:<math>h(x,y)= \sum_\vec{q} a_\vec{q} e^{i\vec{q}\vec{r}}.</math>

Since normal modes are orthogonal, the energy is easily expressible as a sum of terms <math>\propto q^2 |a_\vec{q}|^2</math>. Each term of the sum is quadratic in the amplitude; hence [[equipartition]] holds, according to standard [[statistical mechanics | classical statistical mechanics]], and the mean energy of each mode will be <math>k_B T/2</math>. Surprisingly, this result leads to a '''divergent''' surface (the width of the interface is bound to diverge with its area) (Ref 2). This divergence is nevertheless very mild; even for displacements on the order of meters, the deviation of the surface is comparable to the size of the molecules.
Moreover, the introduction of an external field removes this divergence: the action of gravity is sufficient to keep the width fluctuation on the order
of one molecular diameter for areas larger than about 1 mm2 (Ref. 2).
The action of gravity is taken into account by integrating the potential energy density due to gravity, <math>\rho g z</math> from a reference height to the position of the surface, <math>z=h(x,y)</math>:

:<math>E_\mathrm{g}= \iint dx\, dy\, \int_0^h dz \rho g z = \frac{\rho g}{2} \int dx\, dy\, h^2.</math>

(For simplicity, one neglects the density of the gas above, which is often acceptable; otherwise, instead of the density the difference in densities appears).

Recently, a procedure has been proposed to obtain a molecular intrinsic
surface from simulation data (Ref. 3), the [[intrinsic sampling method]]. The density profiles obtained
from this surface are, in general, quite different from the usual
''mean density profiles''.

==Gravity-capillary waves==
These are ordinary waves excited in an interface, such as ripples on
a water surface. Their dispersion relation reads, for waves on the interface between two fluids of infinite depth:

:<math>
\omega^2=\frac{\rho-\rho'}{\rho+\rho'}gk+\frac{\sigma}{\rho+\rho'}k^3,</math>

where <math>\omega</math> is the angular frequency, <math>g</math> the acceleration due to gravity, <math>\sigma</math> the [[surface tension]], <math>\rho</math> and <math>\rho^'</math> the mass density of the two fluids (<math>\rho > \rho^'</math>) and <math>k</math> the is wavenumber.
===Derivation===
This is a sketch of the derivation of the general dispersion relation, see Ref. 4 for a more detailed description. The problem is unfortunately a bit complex. As Richard Feynman put it (Ref. 6):
<blockquote>
"''...[water waves], which are easily seen by everyone and which are used as an example of waves in elementary courses... are the worst possible example... they have all the complications that waves can have''"
</blockquote>
====Defining the problem====
Three contributions to the energy are involved: the [[surface tension]], gravity, and [[hydrodynamics]]. The parts due to surface tension (again the derivatives are taken to be small) and gravity are exactly as above.
The new contribution involves the [[kinetic energy]] of the fluid:

:<math>T= \frac{\rho}{2} \iint dx\, dy\, \int_{-\infty}^h dz v^2,</math>

where <math>v</math> is the module of the velocity field <math>\vec{v}</math>.
(Again, we are neglecting the flow of the gas above for simplicity.)

====Wave solutions====
Let us suppose the surface of the liquid is described by a traveling plane wave:

:<math>h(x,y,t)=\eta(t)e^{i\vec{q}\vec{r}},</math>

where <math>\eta(t)=\exp[i\omega t]</math> and <math>\vec{q}=(q_x,q_y)</math> is a two dimensional wave number vector, <math>\vec{r}=(x,y)</math> being the horizontal position. We may take <math>\vec{q}=(q,0)</math> without loss of generality:

:<math>h(x,y,t)=\eta(t)e^{i q x}.</math>

In this case it is easy to perform the integrations involved in the expressions for the energies. The
integration over <math>x</math> can taken over a period of oscillation <math>\lambda=2\pi/q</math>, then
multiplied by the number of oscillations in our very large (in principle, infinite) system: <math>L_x / \lambda</math>. The integration over <math>y</math> trivially yields <math>L_y</math>. Performing the integrations, keeping in mind that only the real part of complex numbers is to be taken as physical, one finds:

:<math>E_\mathrm{g} = \frac{A}{2} \frac{\rho g}{2} \eta^2,</math>
:<math>E_\mathrm{st} = \frac{A}{2} \frac{\sigma}{2} q^2 \eta^2,</math>

where <math>A=L_x\times L_y</math> is the area of the system.

To tackle the kinetic energy, suppose the fluid is incompressible and its flow is irrotational (often, sensible approximations) - the flow will then be [[potential flow|potential]]: <math>\vec{v}=\nabla\phi</math>, and <math>\phi</math> is a potential (scalar field) which must satisfy [[Laplace's equation]] <math>\nabla^2\phi=0</math>.
If we try try separation of variables with the potential:

:<math>\phi(x,y,z,t)=\xi(t) f(z) e^{i q x},</math>

with some function of time <math>\xi(t)</math>, and some function of vertical component (height) <math>f(z)</math>.
Laplace's equation then requires on the later

:<math>f''(z)= q^2 f(z) .</math>

This equation can be solved with the proper boundary conditions: first, <math>\vec{v}</math> must vanish well below the surface (in the "deep water" case, which is the one we consider, otherwise a more general relation holds, which is also well known in oceanography). Therefore

:<math>f(z) = a \, \exp(|q| z) </math>,

with some constant <math>a</math>. The less trivial condition is the proper matching between <math>\phi</math> and <math>h</math>: the potential field must correspond to a velocity field that is adjusted to the movement of the surface:

:<math>v_z (z=h) =\partial h/\partial t</math>.

(Actually, this is the linearized version of a more general expression, see below.)

This implies that:

:<math>\xi(t)=\eta(t)'</math>, and <math>f'(z=h) = 1, </math>

so that

:<math>f(z) = \exp( -|q|(h-z))/|q| </math>.

We may now find the velocity field, <math>\vec{v}=\nabla\phi</math>, which shows the well-known circles: the elements of fluid undergo circular motion in the <math>x,z</math> plane, with the circles getting smaller at deeper levels. The displacement of a fluid element is given by <math>\partial\vec{\psi}/\partial t= \vec{v}</math>, and is plotted in Figure 1.

[[Image:Wave_v_field.jpg|300px|thumb|right|Figure 1. Displacement of particle (snapshot)]]

For the kinetic energy, we need
<math>v^2=|\nabla\phi|^2</math>, which is:

:<math>v^2= (\eta')^2 e^{ -2 |q|(h-z)}, </math>

with no dependence on <math>x</math> or <math>y</math>; the other integration provides:

:<math>T= \frac{\rho A }{2|q|} ( \eta' )^2 \int_{-\infty}^h e^{ -2 |q|(h-z)} = \frac{A}{2} \frac{\rho }{2|q|} ( \eta' )^2 .
</math>

The problem is thus specified by just a potential energy involving the square of <math>\eta(t)</math> and a kinetic energy involving the square of its time derivative: a regular [[Harmonic spring approximation|harmonic oscillator]]. In particular:
:<math>E= \frac{A}{2}
\left[
\left( \rho g + {\sigma} q^2 \right) \frac{\eta^2}{2}+
\frac{\rho}{q} \frac{(\eta')^2}{2}
\right].
</math>
Identifying the oscillator's "spring constant" <math>\kappa = \rho g + {\sigma} q^2 </math>, and its "mass"
<math>m= \rho / q</math>, the oscillation frequency must be given by <math>\omega^2=\kappa/m</math>, which results in:

:<math>\omega^2=g q+\frac{\sigma}{\rho} q^3,</math>

the same dispersion as above if <math>\rho'</math> is neglected.

====Alternative derivation====

In Reference 7 the dispersion relation is derived in a somewhat different manner. The same assumptions are made regarding the fluid (it is inviscid, irrotational, and incompressible), so Laplace's Equation is to be satisfied: <math>\nabla^2\phi=0</math>, with <math>\vec{v}=\nabla\phi</math>. The boundary conditions, on the other hand, are sufficient to solve the problem.

One boundary condition is the requirement that the surface of the liquid, defined by <math>z=h(x,y;t)</math> follows the velocity field:
:<math>
\frac{\partial h}{\partial t}+v_x \frac{\partial h}{\partial x}+v_y \frac{\partial h}{\partial y}= v_z .
</math>
A simpler condition follows from linearization: <math> \partial h /\partial t =v_z </math>, as in the previous derivation. There is an additional boundary condition at the bottom of the fluid, which we take here as <math>v_z=0</math> when <math>z\rightarrow -\infty</math>.

In the same fashion as above, we seek surface wave solutions, of the form <math>h(x,y,t)=a e^{i (qx-\omega t)}</math>. We may guess a solution of the form
:<math>\phi=-i\omega h(x,y;t) f(z).</math>

This first condition implies <math>f'(z=h)=1</math>. Together with Laplace's equation, this leads to a function
:<math>f=(1/q) \exp(q(z-h)). </math>
(see Ref 8 for a discussion on when Laplace's equation admits wave solutions.)

The other surface boundary condition is a [[Bernoulli equation]], stating that the pressure just below the surface, <math>p_-</math>, must equal the [[saturation pressure]] of coexistence, minus a contribution due to the surface:
:<math>
p_-=p -
\rho\left[
\frac{\partial \phi}{\partial t}+
\frac{1}{2} v^2+ gh
\right] .
</math>
The linearized condition is
:<math>
\frac{\partial \phi}{\partial t}+gh = \frac{p-p_-}{\rho}
</math>

The connection with the curvature of the surface can be introduced by [[Young's equation]] for the pressure drop across a curved interface, whose linearized form is:
:<math>
p-p_-=\sigma\left(\frac{\partial^2 h}{\partial x^2}+\frac{\partial^2 h}{\partial y^2}\right),
</math>
where <math>\sigma</math> is the surface tension.
The linearized condition is finally
:<math>
\frac{\partial \phi}{\partial t}+gh = \frac{\sigma}{\rho} \left(\frac{\partial^2 h}{\partial x^2}+\frac{\partial^2 h}{\partial y^2}\right).
</math>

This second condition, when applied to the surface wave above, establishes that <math>f(z=h)=(g+\sigma/\rho q^2)/\omega^2</math>.

For the two conditions to hold, <math>1/q</math> must equal <math>(g+\sigma/\rho q^2)/\omega^2</math>, which is precisely the same dispersion relation as the one above.

This derivation makes clear the assumptions introduced. In particular, the linearization will only hold for smooth waves, the ones for which the wave amplitude, <math>a</math> is smaller than the wavelength. Mathematically, the limit is <math>q a \ll 1</math>. For ocean waves, this happens when waves approach the shore and the amplitude grows (in this limit, a bottom boundary condition <math>v_z (z=-H) =0</math> must be employed, and waves are not dispersive, see Ref 7.)
==References==
#[http://dx.doi.org/10.1103/PhysRevLett.15.621 F. P. Buff, R. A. Lovett, and F. H. Stillinger, Jr. "Interfacial density profile for fluids in the critical region" Physical Review Letters '''15''' pp. 621-623 (1965)]
#J. S. Rowlinson and B. Widom "Molecular Theory of Capillarity". Dover 2002 (originally: Oxford University Press 1982) ISBN 0486425444
#[http://dx.doi.org/10.1103/PhysRevLett.91.166103 E. Chacón and P. Tarazona "Intrinsic profiles beyond the capillary wave theory: A Monte Carlo study", Physical Review Letters '''91''' 166103 (2003)]
#Samuel A. Safran "Statistical thermodynamics of surfaces, interfaces, and membranes" Addison-Wesley 1994 ISBN 9780813340791
#[http://dx.doi.org/10.1103/PhysRevLett.99.196101 P. Tarazona, R. Checa, and E. Chacón "Critical Analysis of the Density Functional Theory Prediction of Enhanced Capillary Waves", Physical Review Letters '''99''' 196101 (2007)]
#R.P. Feynman, R.B. Leighton, and M. Sands "The Feynman lectures on physics" Addison-Wesley 1963. Section 51-4. ISBN 0201021153
#[http://dx.doi.org/10.1006/rwos.2001.0129 W.K. Melville "Surface, gravity and capillary waves"], in [http://www.sciencedirect.com/science/referenceworks/9780122274305 "Encyclopedia of Ocean Sciences"], Eds: Steve A. Thorpe and Karl K. Turekian. Elsevier 2001, page 2916. ISBN 978-0-12-227430-5
#[http://dx.doi.org/10.1088/0143-0807/25/1/014 F Behroozi "Fluid viscosity and the attenuation of surface waves: a derivation based on conservation of energy", European Journal of Physics '''25''' 115 (2004)]
==External links==
*[http://en.wikipedia.org/wiki/Capillary_wave Capillary wave entry in Wikipedia]
*[http://en.wikipedia.org/wiki/Thermal_capillary_wave Thermal capillary wave entry in Wikipedia]
[[Category: Classical thermodynamics ]]

Mean field models

2010-05-03T14:28:10Z

Dduque: Added link to Curie's_law

A '''mean field model''', or a '''mean field solution''' of a model, is an approximation to the actual solution of a model in statistical physics. The model is made exactly solvable by treating the effect of all other particles on a given one as a ''mean field'' (hence its name). It appear in different forms and different contexts, but all mean field models have this feature in common.

==Mean field solution of the Ising model==

A well-known mean field solution of the [[Ising model]], known as the ''Bragg-Williams approximation'' goes as follows.
From the original Hamiltonian,
:<math> U = - J \sum_i^N S_i \sum_{<j>} S_j , </math>
suppose we may approximate
:<math> \sum_{<j>} S_j \approx n \bar{s}, </math>
where <math>n</math> is the number of neighbors of site <math>i</math> (e.g. 4 in a 2-D square lattice), and <math>\bar{s}</math> is the (unknown) magnetization:
:<math> \bar{s}=\frac{1}{N} \sum_i S_i . </math>

Therefore, the Hamiltonian turns to
:<math> U = - J n \sum_i^N S_i \bar{s} , </math>
as in the regular Langevin theory of magnetism (see [[Curie's_law]]): the spins are independent, but coupled to a constant field of strength
:<math>H= J n \bar{s}.</math>
The magnetization of the Langevin theory is
:<math> \bar{s} = \tanh( H/k_B T ). </math>
Therefore:
:<math> \bar{s} = \tanh(J n\bar{s}/k_B T). </math>

This is a '''self-consistent''' expression for <math>\bar{s}</math>. There exists a critical temperature, defined by
:<math>k_B T_c= J n .</math>
At temperatures higher than this value the only solution is <math>\bar{s}=0</math>. Below it, however, this solution becomes unstable
(it corresponds to a maximum in energy), whereas two others are stable. Slightly below <math>T_c</math>,
:<math>\bar{s} =\pm\sqrt{3\left(1 - \frac{T}{T_c}\right)}. </math>

==General discussion==
The solution obtained shares a number of features with any other mean field approximation:
*It largely ignores geometry, which may be important in some cases. In particular, it reduces the lattice details to just the number of neighbours.
*As a consequence, it may predict phase transitions where none are found: the [[1-dimensional_Ising_model|1-D ising model]] <math>n=2</math> is known to lack any phase transition (at finite temperature)
*In general, the theory ''underestimates fluctuations''
*It also leads to ''classical critical exponents'', like the <math>\left(1 - \frac{T}{T_c}\right)^{1/2}</math> decay above. In 3-D, the magnetization follows a power law with a different exponent.
*Nevertheless, above a certain space dimension the critical exponents are correct. This dimension is 4 for the Ising model, as predicted by a self-consistency requirement due to E.M. Lipfshitz (similar ones are due to Peierls and L.D. Landau)

Mean field models

2010-05-03T14:26:32Z

Dduque: Completed solution, section added.

A '''mean field model''', or a '''mean field solution''' of a model, is an approximation to the actual solution of a model in statistical physics. The model is made exactly solvable by treating the effect of all other particles on a given one as a ''mean field'' (hence its name). It appear in different forms and different contexts, but all mean field models have this feature in common.

==Mean field solution of the Ising model==

A well-known mean field solution of the [[Ising model]], known as the ''Bragg-Williams approximation'' goes as follows.
From the original Hamiltonian,
:<math> U = - J \sum_i^N S_i \sum_{<j>} S_j , </math>
suppose we may approximate
:<math> \sum_{<j>} S_j \approx n \bar{s}, </math>
where <math>n</math> is the number of neighbors of site <math>i</math> (e.g. 4 in a 2-D square lattice), and <math>\bar{s}</math> is the (unknown) magnetization:
:<math> \bar{s}=\frac{1}{N} \sum_i S_i . </math>

Therefore, the Hamiltonian turns to
:<math> U = - J n \sum_i^N S_i \bar{s} , </math>
as in the regular Langevin theory of magnetism: the spins are independent, but coupled to a constant field of strength
:<math>H= J n \bar{s}.</math>
The magnetization of the Langevin theory is
:<math> \bar{s} = \tanh( H/k_B T ). </math>
Therefore:
:<math> \bar{s} = \tanh(J n\bar{s}/k_B T). </math>

This is a '''self-consistent''' expression for <math>\bar{s}</math>. There exists a critical temperature, defined by
:<math>k_B T_c= J n .</math>
At temperatures higher than this value the only solution is <math>\bar{s}=0</math>. Below it, however, this solution becomes unstable
(it corresponds to a maximum in energy), whereas two others are stable. Slightly below <math>T_c</math>,
:<math>\bar{s} =\pm\sqrt{3\left(1 - \frac{T}{T_c}\right)}. </math>

==General discussion==
The solution obtained shares a number of features with any other mean field approximation:
*It largely ignores geometry, which may be important in some cases. In particular, it reduces the lattice details to just the number of neighbours.
*As a consequence, it may predict phase transitions where none are found: the [[1-dimensional_Ising_model|1-D ising model]] <math>n=2</math> is known to lack any phase transition (at finite temperature)
*In general, the theory ''underestimates fluctuations''
*It also leads to ''classical critical exponents'', like the <math>\left(1 - \frac{T}{T_c}\right)^{1/2}</math> decay above. In 3-D, the magnetization follows a power law with a different exponent.
*Nevertheless, above a certain space dimension the critical exponents are correct. This dimension is 4 for the Ising model, as predicted by a self-consistency requirement due to E.M. Lipfshitz (similar ones are due to Peierls and L.D. Landau)

Mean field models

2010-05-03T14:17:02Z

Dduque: /* Mean field solution of the Ising model */

Mean field models

2010-04-29T10:34:36Z

Dduque: I can't believe there was not an entry about this... work in progress

Ising model

2010-04-29T10:26:30Z

Dduque: Added link to Mean field models

The '''Ising model''' <ref>[http://dx.doi.org/10.1007/BF02980577 Ernst Ising "Beitrag zur Theorie des Ferromagnetismus", Zeitschrift für Physik A Hadrons and Nuclei '''31''' pp. 253-258 (1925)]</ref> (also known as the '''Lenz-Ising''' model) is commonly defined over an ordered lattice.
Each site of the lattice can adopt two states, <math>S \in \{-1, +1 \}</math>. Note that sometimes these states are referred to as ''spins'' and the values are referred to as ''down'' and ''up'' respectively.
The energy of the system is the sum of pair interactions
between nearest neighbors.

:<math> \frac{U}{k_B T} = - K \sum_{\langle ij \rangle} S_i S_j </math>

where <math>k_B</math> is the [[Boltzmann constant]], <math>T</math> is the [[temperature]], <math> \langle ij \rangle </math> indicates that the sum is performed over nearest neighbors, and
<math> S_i </math> indicates the state of the i-th site, and <math> K </math> is the coupling constant.

For a detailed and very readable history of the Lenz-Ising model see the following references:<ref>[http://dx.doi.org/10.1103/RevModPhys.39.883 S. G. Brush "History of the Lenz-Ising Model", Reviews of Modern Physics '''39''' pp. 883-893 (1967)]</ref>
<ref>[http://dx.doi.org/10.1007/s00407-004-0088-3 Martin Niss "History of the Lenz-Ising Model 1920-1950: From Ferromagnetic to Cooperative Phenomena", Archive for History of Exact Sciences '''59''' pp. 267-318 (2005)]</ref>
<ref>[http://dx.doi.org/10.1007/s00407-008-0039-5 Martin Niss "History of the Lenz–Ising Model 1950–1965: from irrelevance to relevance", Archive for History of Exact Sciences '''63''' pp. 243-287 (2009)]</ref>.
==1-dimensional Ising model==
:''Main article: [[1-dimensional Ising model]]''
The 1-dimensional Ising model has an exact solution.

==2-dimensional Ising model==
The 2-dimensional Ising model was solved by [[Lars Onsager]] in 1944
<ref>[http://dx.doi.org/10.1103/PhysRev.65.117 Lars Onsager "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition", Physical Review '''65''' pp. 117 - 149 (1944)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRev.88.1332 M. Kac and J. C. Ward "A Combinatorial Solution of the Two-Dimensional Ising Model", Physical Review '''88''' pp. 1332-1337 (1952)]</ref>
<ref>Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 7 (freely available [http://tpsrv.anu.edu.au/Members/baxter/book/Exactly.pdf pdf])</ref>
after [[Rudolf Peierls]] had previously shown that, contrary to the one-dimensional case, the two-dimensional model must have a phase transition
<ref>[http://dx.doi.org/10.1017/S0305004100019174 Rudolf Peierls "On Ising's model of ferromagnetism", Mathematical Proceedings of the Cambridge Philosophical Society '''32''' pp. 477-481 (1936)]</ref> <ref>[http://dx.doi.org/10.1103/PhysRev.136.A437 Robert B. Griffiths "Peierls Proof of Spontaneous Magnetization in a Two-Dimensional Ising Ferromagnet", Physical Review A '''136''' pp. 437-439 (1964)]</ref>.

==3-dimensional Ising model==
Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice
<ref>[http://www.sandia.gov/LabNews/LN04-21-00/sorin_story.html Three-dimensional proof for Ising model impossible, Sandia researcher claims to have shown]</ref>
<ref>[http://dx.doi.org/10.1145/335305.335316 Sorin Istrail "Statistical mechanics, three-dimensionality and NP-completeness: I. Universality of intracatability for the partition function of the Ising model across non-planar surfaces", Proceedings of the thirty-second annual ACM symposium on Theory of computing pp. 87-96 (2000)]</ref>
==ANNNI model==
The '''axial next-nearest neighbour Ising''' (ANNNI) model <ref>[http://dx.doi.org/10.1016/0370-1573(88)90140-8 Walter Selke "The ANNNI model — Theoretical analysis and experimental application", Physics Reports '''170''' pp. 213-264 (1988)]</ref> is used to study alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes.
==See also==
*[[Critical exponents]]
*[[Potts model]]
*[[Mean field models]]

==References==
<references/>
[[Category: Models]]

Statistical mechanics

2010-04-29T10:25:24Z

Dduque: Added link to Mean field models

{{columns-list|3|
==B==
*[[BBGKY hierarchy]]
*[[Boltzmann average]]
*[[Boltzmann constant]]
*[[Boltzmann distribution]]
*[[Boltzmann factor]]
*[[Born-Green equation]]
*[[Brownian motion]]

==C==
*[[Canonical ensemble]]
*[[Chemical potential]]
*[[Compressibility equation]]
*[[Critical exponents]]
*[[Critical points]]
*[[Curie's law]]

==D==
*[[Darwin-Fowler method]]
*[[de Broglie thermal wavelength]]

==E==
*[[Energy equation]]
*[[Entropy]]
*[[Ensembles in thermostatistics]]
*[[Ergodic hypothesis]]

==F==
*[[Fermi-Pasta-Ulam experiment]]
*[[Fisher–Ruelle stability criteria]]
*[[Fisher-Widom line]]
*[[Four-body function]]

==G==
*[[Gibbs distribution]]
*[[Gibbs ensemble]]
*[[Gibbs measures]]
*[[Gibbs paradox]]
*[[Goldstone modes]]
*[[Grand canonical ensemble]]

==I==
*[[Information theory]]
*[[Internal energy]]
*[[Intermolecular pair potential]]
*[[Isoenthalpic–isobaric ensemble]]
*[[Isothermal-isobaric ensemble]]
==J==
*[[Joule-Thomson effect]]
==K==
*[[Kirkwood superposition approximation]]
*[[Kolmogorov–Arnold–Moser theorem]]

==L==
*[[Law of corresponding states]]
*[[Liouville's theorem]]
*[[Loschmidt's paradox]] (Umkehreinwand)
*[[Lyapunov exponents]]

==M==
*[[Maxwell velocity distribution]]
*[[Maxwell's demon]]
*[[Mean field models]]
*[[Microcanonical ensemble]]
*[[Microstate]]
*[[Mixing systems]]

==P==
*[[Pair distribution function]]
*[[Parrondo's paradox]]
*[[Partition function]]
*[[Phase space]]
*[[Phase transitions]]
*[[Poincaré theorem]]
*[[Pressure]]
*[[Pressure equation]] (aka. virial equation)

==Q==
*[[Quantum statistics]]
==R==
*[[Radial distribution function]]
==S==
*[[Semi-grand ensembles]]
*[[Stochastic transition]]
*[[Structure factor]]
*[[Surface tension]]

==T==
*[[Temperature]]
*[[Thermodynamic limit]]
*[[Transfer matrices]]
==Y==
*[[Yang-Yang anomaly]]
==Z==
*[[Zermelo’s paradox]] (Wiederkehreinwand)
}}
==General reading==
'''Books'''
* Donald A. McQuarrie "Statistical Mechanics", University Science Books (1984) (Re-published 2000) ISBN 978-1-891389-15-3
* L. D. Landau and E. M. Lifshitz "Statistical Physics", Course of Theoretical Physics volume 5 Part 1 3rd Edition (1984) ISBN 0750633727
* Terrell L. Hill "Statistical Mechanics: Principles and Selected Applications" (1956) ISBN 0486653900
* Terrell L. Hill "An Introduction to Statistical Thermodynamics" (1986) ISBN 0486652424
'''Papers'''
*[http://dx.doi.org/10.1103/RevModPhys.27.289 D. Ter Haar "Foundations of Statistical Mechanics", Reviews of Modern Physics '''27''' pp. 289 - 338 (1955)]
*[http://dx.doi.org/10.1088/0034-4885/42/12/002 O. Penrose "Foundations of statistical mechanics", Reports on Progress in Physics '''42''' pp. 1937-2006 (1979)]
*[http://dx.doi.org/10.1126/science.177.4047.393 P. W. Anderson "More Is Different", Science '''177''' pp. 393-396 (1972)]
*[http://dx.doi.org/10.1080/00268970600965835 J. S. Rowlinson "The evolution of some statistical mechanical ideas", Molecular Physics '''104''' pp. 3399 - 3410 (2006)]
*[http://dx.doi.org/10.1016/S0378-4371(98)00517-2 Elliott H. Lieb "Some problems in statistical mechanics that I would like to see solved", Physica A '''263''' pp. 491-499 (1999)]
*[http://www.pro-physik.de/Phy/pdfstart.do?mid=3&articleid=25753&recordid=25793 Joel L. Lebowitz "Emergent Phenomena", Physik Journal '''6''' #9 pp. 41-46 (2007)]
[[category: statistical mechanics]]
__NOTOC__

Beeman's algorithm

2010-04-19T11:01:10Z

Dduque: Interesting: same positions as Verlet

'''Beeman's algorithm''' <ref>[http://dx.doi.org/10.1016/0021-9991(76)90059-0 D. Beeman "Some multistep methods for use in molecular dynamics calculations", Journal of Computational Physics '''20''' pp. 130-139 (1976)]</ref> is is a method for [[Integrators for molecular dynamics |numerically integrating ordinary differential equations]], generally position and velocity, which is closely related to Verlet integration.

In its standard form, it produces the same trajectories as the Verlet algorithm, but the velocities are more accurate:

:<math>x(t+\Delta t) = x(t) + v(t) \Delta t + \left(\frac{2}{3}a(t) - \frac{1}{6} a(t - \Delta t) \right)\Delta t^2 + O( \Delta t^4) </math>
:<math>v(t + \Delta t) = v(t) + \left(\frac{1}{3}a(t + \Delta t) + \frac{5}{6}a(t) - \frac{1}{6}a(t - \Delta t) \right) \Delta t + O(\Delta t^3)</math>

where ''x'' is the position, ''v'' is the velocity, ''a'' is the acceleration, ''t'' is time, and <math>\Delta t</math> is the [[Time step|time-step]].

A predictor-corrector variant is useful when the forces are velocity-dependent:

:<math> x(t+\Delta t) = x(t) + v(t) \Delta t + \frac{2}{3}a(t) \Delta t^2 - \frac{1}{6} a(t - \Delta t) \Delta t^2 + O( \Delta t^4).</math>

The velocities at time <math>t =t + \Delta t</math> are then calculated from the positions.

:<math> v(t + \Delta t)_{(\mathrm{predicted})} = v(t) + \frac{3}{2}a(t) \Delta t - \frac{1}{2}a(t - \Delta t) \Delta t + O( \Delta t^3)</math>

The accelerations at time <math>t =t + \Delta t</math> are then calculated from the positions and predicted velocities.

:<math> v(t + \Delta t)_{(\mathrm{corrected})} = v(t) + \frac{1}{3}a(t + \Delta t) \Delta t + \frac{5}{6}a(t) \Delta t - \frac{1}{6}a(t - \Delta t) \Delta t + O( \Delta t^3) </math>

==See also==
*[[Velocity Verlet algorithm]]

==References==
<references/>
==External links==
*[http://en.wikipedia.org/wiki/Beeman%27s_algorithm Beeman's algorithm entry on wikipedia]
[[category: Molecular dynamics]]

Beeman's algorithm

2010-04-17T13:35:31Z

Dduque: Link to wikipedia

'''Beeman's algorithm''' is is a method for numerically integrating ordinary differential equations, generally position and velocity, which is closely related to Verlet integration.

:<math>x(t+\Delta t) = x(t) + v(t) \Delta t + (\frac{2}{3}a(t) - \frac{1}{6} a(t - \Delta t) )\Delta t^2 + O( \Delta t^4) </math>
:<math>v(t + \Delta t) = v(t) + (\frac{1}{3}a(t + \Delta t) + \frac{5}{6}a(t) - \frac{1}{6}a(t - \Delta t)) \Delta t + O(\Delta t^3)</math>

where ''x'' is the position, ''v'' is the velocity, ''a'' is the acceleration, ''t'' is time, and ''\Delta t'' is the time-step.

A predictor-corrector variant is useful when the forces are velocity-dependent:

:<math> x(t+\Delta t) = x(t) + v(t) \Delta t + \frac{2}{3}a(t) \Delta t^2 - \frac{1}{6} a(t - \Delta t) \Delta t^2 + O( \Delta t^4).</math>

The velocities at time <math>t =t + \Delta t</math> are then calculated from the positions.

:<math> v(t + \Delta t) (predicted) = v(t) + \frac{3}{2}a(t) \Delta t - \frac{1}{2}a(t - \Delta t) \Delta t + O( \Delta t^3)</math>

The accelerations at time <math>t =t + \Delta t</math> are then calculated from the positions and predicted velocities.

:<math> v(t + \Delta t) (corrected) = v(t) + \frac{1}{3}a(t + \Delta t) \Delta t + \frac{5}{6}a(t) \Delta t - \frac{1}{6}a(t - \Delta t) \Delta t + O( \Delta t^3) </math>

==See also==
*[[Velocity Verlet algorithm]]

==References==
*[http://en.wikipedia.org/wiki/Beeman%27s_algorithm Beeman's algorithm entry on wikipedia]
[[category: Molecular dynamics]]

Beeman's algorithm

2010-04-17T13:07:22Z

Dduque: common factor

Beeman's algorithm

2010-04-17T10:16:17Z

Dduque: New page: '''Beeman's algorithm''' is is a method for numerically integrating ordinary differential equations, generally position and velocity, which is closely related to Verlet integration. :<mat...

'''Beeman's algorithm''' is is a method for numerically integrating ordinary differential equations, generally position and velocity, which is closely related to Verlet integration.

:<math>x(t+\Delta t) = x(t) + v(t) \Delta t + \frac{2}{3}a(t) \Delta t^2 - \frac{1}{6} a(t - \Delta t) \Delta t^2 + O( \Delta t^4) </math>
:<math>v(t + \Delta t) = v(t) + \frac{1}{3}a(t + \Delta t) \Delta t + \frac{5}{6}a(t) \Delta t - \frac{1}{6}a(t - \Delta t) \Delta t + O(\Delta t^3)</math>

where ''x'' is the position, ''v'' is the velocity, ''a'' is the acceleration, ''t'' is time, and ''\Delta t'' is the time-step.

A predictor-corrector variant is useful when the forces are velocity-dependent:

:<math> x(t+\Delta t) = x(t) + v(t) \Delta t + \frac{2}{3}a(t) \Delta t^2 - \frac{1}{6} a(t - \Delta t) \Delta t^2 + O( \Delta t^4).</math>

The velocities at time <math>t =t + \Delta t</math> are then calculated from the positions.

:<math> v(t + \Delta t) (predicted) = v(t) + \frac{3}{2}a(t) \Delta t - \frac{1}{2}a(t - \Delta t) \Delta t + O( \Delta t^3)</math>

The accelerations at time <math>t =t + \Delta t</math> are then calculated from the positions and predicted velocities.

:<math> v(t + \Delta t) (corrected) = v(t) + \frac{1}{3}a(t + \Delta t) \Delta t + \frac{5}{6}a(t) \Delta t - \frac{1}{6}a(t - \Delta t) \Delta t + O( \Delta t^3) </math>

==See also==
*[[Velocity Verlet algorithm]]
[[category: Molecular dynamics]]