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Path integral formulation

2011-05-12T09:51:54Z

RamiroRoberts:

The '''path integral formulation''', here from the [[statistical mechanics | statistical mechanical]] point of view, is an elegant method by which [[quantum mechanics | quantum mechanical]] contributions can be incorporated
within a classical [[Computer simulation techniques |simulation]] using Feynman path integrals (see the [[Path integral formulation#Additional reading|additional reading ]] section). Such simulations are particularly applicable to light atoms and molecules such as [[hydrogen]], [[helium]], [[neon]] and [[argon]], as well as quantum rotators such as [[methane]] and hydrogen-bonded systems such as [[water]]. From a more idealised point of view path integrals are often used to study [[quantum hard spheres]].
==Principles==
In the path integral formulation the canonical [[partition function]] (in one dimension) is written as
(<ref name="Berne">[http://dx.doi.org/10.1146/annurev.pc.37.100186.002153 B. J. Berne and D. Thirumalai "On the Simulation of Quantum Systems: Path Integral Methods", Annual Review of Physical Chemistry '''37''' pp. 401-424 (1986)]</ref> Eq. 1)
:<math>Q(\beta, V)= \int {\mathrm d} x_1 \int_{x_1}^{x_1} Dx(\tau)e^{-S[x(\tau)]}</math>
where <math>S[x(\tau)]</math> is the Euclidian action, given by (<ref name="Berne"> </ref> Eq. 2)
:<math>S[x(\tau)] = \int_0^{\beta \hbar} H(x(\tau))</math>
where <math>x(\tau)</math> is the path in time <math>\tau</math> and <math>H</math> is the [[Hamiltonian]].
This leads to (<ref name="Berne"> </ref> Eq. 3)
:<math>Q_P = \left( \frac{mP}{2 \pi \beta \hbar^2} \right)^{P/2} \int ... \int {\mathrm d}x_1... {\mathrm d}x_P e^{-\beta \Phi_P (x_1...x_P;\beta)}</math>
where the Euclidean time is discretised in units of
:<math>\varepsilon = \frac{\beta \hbar}{P}, P \in {\mathbb Z}</math>
:<math>x_t = x(t \beta \hbar/P)</math>
:<math>x_{P+1}=x_1</math>
and (<ref name="Berne"> </ref> Eq. 4)
:<math>\Phi_P (x_1...x_P;\beta)= \frac{mP}{2\beta^2 \hbar^2} \sum_{t=1}^P (x_t - x_{t+1})^2 + \frac{1}{P} \sum_{t=1}^P V(x_t)</math>.

where <math>P</math> is the Trotter number. In the Trotter limit, where <math>P \rightarrow \infty</math> these equations become exact. In the case where <math>P=1</math> these equations revert to a classical simulation. It has long been recognised that there is an isomorphism between this discretised quantum mechanical description, and the classical [[statistical mechanics]] of polyatomic fluids, in particular flexible ring molecules<ref>[http://dx.doi.org/10.1063/1.441588 David Chandler and Peter G. Wolynes "Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids", Journal of Chemical Physics '''74''' pp. 4078-4095 (1981)]</ref>, due to the periodic boundary conditions in imaginary time. It can be seen from the first term of the above equation that each particle <math>x_t</math> interacts with is neighbours <math>x_{t-1}</math> and <math>x_{t+1}</math> via a harmonic spring. The second term provides the internal potential energy.
The following is a schematic for the interaction between two (purple) atoms. Here we show the atoms having five Trotter slices, forming what can be thought of as a "ring polymer molecule". The harmonic springs between Trotter slices are in red, and white/green bonds represent the [[intermolecular pair potential]].<br>
[[Image:5bead_pathIntegral.png|500px]]<br>
In three dimensions one has the ''density operator''

:<math>\hat{\rho} (\beta) = \exp\left[ -\beta \hat{H} \right]</math>

which thanks to the [[Trotter formula]] we can tease out <math>\exp \left[ -\beta (U_{\mathrm {spring}}+ U_{\mathrm{internal}} ) \right]</math>, where

:<math>U_{\mathrm {spring}} = \frac{mP}{2\beta^2 \hbar^2} \sum_{t=1}^P | \mathbf{r}_t - \mathbf{r}_{t+1} |^2</math>

and

:<math>U_{\mathrm{internal}}= \frac{1}{P} \sum_{t=1}^P V(\mathbf{r}_t)</math>

The [[internal energy]] is given by

:<math>\langle U \rangle = \frac{3NP}{2\beta}- \langle U_{\mathrm {spring}} \rangle + \langle U_{\mathrm{internal}} \rangle </math>

The average kinetic energy is known as the ''primitive estimator'', i.e.

:<math>\langle K_P \rangle = \frac{3NP}{2\beta}- \langle U_{\mathrm {spring}} \rangle </math>

==Harmonic oscillator==
The density matrix for a harmonic oscillator is given by (<ref>R. P. Feynman and A. R. Hibbs "Path-integrals and Quantum Mechanics", McGraw-Hill, New York (1965) ISBN 0-07-020650-3</ref> Eq. 10-44)

:<math>\rho(x',x)= \sqrt{ \frac{m \omega}{2 \pi \hbar \sinh \omega \beta \hbar} } \exp \left( - \frac{m \omega}{2 \hbar (\sinh \omega \beta \hbar)^2 } \left( (x^2 + x'^2 ) \cosh \omega \beta \hbar - 2xx'\right)\right)</math>

'''Related reading'''
*[http://dx.doi.org/10.1119/1.18910 Barry R. Holstein "The harmonic oscillator propagator", American Journal of Physics '''66''' pp. 583-589 (1998)]
*[http://dx.doi.org/10.1119/1.1715108 L. Moriconi "An elementary derivation of the harmonic oscillator propagator", American Journal of Physics '''72''' pp. 1258-1259 (2004)]
==Wick rotation and imaginary time==
One can identify the [[Temperature#Inverse_temperature | inverse temperature]], <math>\beta</math> with an imaginary time <math>it/\hbar</math> (see <ref>M. J. Gillan "The path-integral simulation of quantum systems" in "Computer Modelling of Fluids Polymers and Solids" eds. C. R. A. Catlow, S. C. Parker and M. P. Allen, NATO ASI Series C '''293''' pp. 155-188 (1990) ISBN 978-0-7923-0549-1</ref> § 2.4).
*[http://dx.doi.org/10.1103/PhysRev.96.1124 G. C. Wick "Properties of Bethe-Salpeter Wave Functions", Physical Review '''96''' pp. 1124-1134 (1954)]

==Rotational degrees of freedom==
In the case of systems having (<math>d</math>) rotational [[degree of freedom | degrees of freedom]] the [[Hamiltonian]] can be written in the form (<ref name="Marx">[http://dx.doi.org/10.1088/0953-8984/11/11/003 Dominik Marx and Martin H Müser "Path integral simulations of rotors: theory and applications", Journal of Physics: Condensed Matter '''11''' pp. R117-R155 (1999)]</ref> Eq. 2.1):
:<math>\hat{H} = \hat{T}^{\mathrm {translational}} + \hat{T}^{\mathrm {rotational}}+ \hat{V}</math>

where the rotational part of the kinetic energy operator is given by (<ref name="Marx"> </ref> Eq. 2.2)

:<math>T^{\mathrm {rotational}} = \sum_{i=1}^{d^{\mathrm {rotational}}} \frac{\hat{L}_i^2}{2\Theta_{ii}}</math>

where <math>\hat{L}_i</math> are the components of the angular momentum operator, and <math>\Theta_{ii}</math> are the moments of inertia.
==Rigid rotators==
:''Main article: [[Rigid top propagator]]''
==Computer simulation techniques==
The following are a number of commonly used [[computer simulation techniques]] that make use of the path integral formulation applied to phases of condensed matter
====Path integral Monte Carlo====
Path integral Monte Carlo (PIMC)
*[http://dx.doi.org/10.1063/1.437829 J. A. Barker "A quantum-statistical Monte Carlo method; path integrals with boundary conditions", Journal of Chemical Physics '''70''' pp. 2914- (1979)]
====Path integral molecular dynamics====
Path integral molecular dynamics (PIMD)
*[http://dx.doi.org/10.1063/1.446740 M. Parrinello and A. Rahman "Study of an F center in molten KCl", Journal of Chemical Physics '''80''' pp. 860- (1984)]
====Centroid molecular dynamics====
Centroid molecular dynamics (CMD)
*[http://dx.doi.org/10.1063/1.467175 Jianshu Cao and Gregory A. Voth "The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties", Journal of Chemical Physics '''100''' pp. 5093-5105 (1994)]
*[http://dx.doi.org/10.1063/1.467176 Jianshu Cao and Gregory A. Voth "The formulation of quantum statistical mechanics based on the Feynman path centroid density. II. Dynamical properties", Journal of Chemical Physics '''100''' pp. 5106- (1994)]
*[http://dx.doi.org/10.1063/1.479515 Seogjoo Jang and Gregory A. Voth "A derivation of centroid molecular dynamics and other approximate time evolution methods for path integral centroid variables", Journal of Chemical Physics '''111''' pp. 2371- (1999)]
*[http://dx.doi.org/10.1063/1.479666 Rafael Ramírez and Telesforo López-Ciudad "The Schrödinger formulation of the Feynman path centroid density", Journal of Chemical Physics '''111''' pp. 3339-3348 (1999)]
*[http://dx.doi.org/10.1063/1.3484490 E. A. Polyakov, A. P. Lyubartsev, and P. N. Vorontsov-Velyaminov "Centroid molecular dynamics: Comparison with exact results for model systems", Journal of Chemical Physics '''133''' 194103 (2010)]

====Ring polymer molecular dynamics====
Ring polymer molecular dynamics (RPMD)
*[http://dx.doi.org/10.1063/1.1777575 Ian R. Craig and David E. Manolopoulos "Quantum statistics and classical mechanics: Real time correlation functions from ring polymer molecular dynamics", Journal of Chemical Physics '''121''' pp. 3368- (2004)]
*[http://dx.doi.org/10.1063/1.2357599 Bastiaan J. Braams and David E. Manolopoulos "On the short-time limit of ring polymer molecular dynamics", Journal of Chemical Physics '''125''' 124105 (2006)]
'''Contraction scheme'''
*[http://dx.doi.org/10.1063/1.2953308 Thomas E. Markland and David E. Manolopoulos "An efficient ring polymer contraction scheme for imaginary time path integral simulations", Journal of Chemical Physics '''129''' 024105 (2008)]
*[http://dx.doi.org/10.1016/j.cplett.2008.09.019 Thomas E. Markland and David E. Manolopoulos "A refined ring polymer contraction scheme for systems with electrostatic interactions" Chemical Physics Letters '''464''' pp. 256-261 (2008)]

====Normal mode PIMD====

====Grand canonical Monte Carlo====
A path integral version of the [[Widom test-particle method]] for [[grand canonical Monte Carlo]] simulations:
*[http://dx.doi.org/10.1063/1.474874 Qinyu Wang, J. Karl Johnson and Jeremy Q. Broughton "Path integral grand canonical Monte Carlo", Journal of Chemical Physics '''107''' pp. 5108-5117 (1997)]

==Applications==
*[http://dx.doi.org/10.1063/1.470898 Jianshu Cao and Gregory A. Voth "Semiclassical approximations to quantum dynamical time correlation functions", Journal of Chemical Physics '''104''' pp. 273-285 (1996)]
*[http://dx.doi.org/10.1063/1.1316105 C. Chakravarty and R. M. Lynden-Bell "Landau free energy curves for melting of quantum solids", Journal of Chemical Physics '''113''' pp. 9239-9247 (2000)]
==References==
<references/>
==Additional reading==
* P. A. M. Dirac "The Lagrangian in Quantum Mechanics", Physikalische Zeitschrift der Sowjetunion '''3''' pp. 64-72 (1933)
*R. P. Feynman "Statistical Mechanics", Benjamin, Reading, Massachusetts, (1972) ISBN 0-201-36076-4 Chapter 3.
*[http://dx.doi.org/10.1143/JPSJ.35.980 Tohru Morita "Solution of the Bloch Equation for Many-Particle Systems in Terms of the Path Integral", Journal of the Physical Society of Japan '''35''' pp. 980-984 (1973)]
*[http://dx.doi.org/10.1016/0370-1573(75)90030-7 F. W. Wiegel "Path integral methods in statistical mechanics", Physics Reports '''16''' pp. 57-114 (1975)]
*[http://dx.doi.org/10.1063/1.437829 J. A. Barker "A quantum-statistical Monte Carlo method; path integrals with boundary conditions", Journal of Chemical Physics '''70''' pp. 2914-2918 (1979)]
*[http://dx.doi.org/10.1103/RevModPhys.67.279 D. M. Ceperley "Path integrals in the theory of condensed helium", Reviews of Modern Physics '''67''' 279 - 355 (1995)]
*[http://dx.doi.org/10.1080/014423597230190 Charusita Chakravarty "Path integral simulations of atomic and molecular systems", International Reviews in Physical Chemistry '''16''' pp. 421-444 (1997)]

==External links==
*[http://www.smac.lps.ens.fr/index.php/Programs_Chapter_3:_Density_matrices_and_path_integrals Density matrices and path integrals] computer code on SMAC-wiki.
[[Category: Monte Carlo]]
[[category: Quantum mechanics]]
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Capillary waves

2011-05-12T09:51:30Z

RamiroRoberts:

==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 mm<sup>2</sup> (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 ]]
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