Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

4.10: Appendix I- Additional Examples

( \newcommand{\kernel}{\mathrm{null}\,}\)




























































































































































































































































































































\( \newcommand\Dalpha

ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[1], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Dbeta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[2], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Dgamma
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[3], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Ddelta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[4], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Depsilon
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[5], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Dvarepsilon
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[6], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Dzeta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[7], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Deta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[8], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Dtheta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[9], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Dvartheta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[10], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Diota
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[11], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Dkappa
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[12], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Dlambda
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[13], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)





\( \newcommand\Dvarpi
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[14], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)











\( \newcommand\DGamma
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[15], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\DDelta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[16], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\DTheta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[17], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)









































































\( \newcommand\Vmu
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[18], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Vnu
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[19], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Vxi
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[20], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Vom
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[21], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Vpi
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[22], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Vvarpi
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[23], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Vrho
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[24], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Vvarrho
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[25], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Vsigma
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[26], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Vvarsigma
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[27], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Vtau
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[28], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Vupsilon
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[29], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Vphi
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[30], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Vvarphi
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[31], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Vchi
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[32], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Vpsi
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[33], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\Vomega
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[34], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\VGamma
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[35], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)
\( \newcommand\VDelta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[36], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)

















\newcommand\BI{\mib I}}










































\)










































































\newcommand { M}

























\newcommand { m}














































}


















\( \newcommand\tcb{\textcolor{blue}\)
\( \newcommand\tcr{\textcolor{red}\)



































1$#1_$






















































































\newcommand\SZ{\textsf Z}} \( \newcommand\kFd{k\ns_{\RF\dar}\)

\newcommand\mutB{\tilde\mu}\ns_\ssr{B}



\( \newcommand\xhihOZ
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/span[1], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)



\( \newcommand\labar
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/span[2], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/04:_Statistical_Ensembles/4.10:_Appendix_I-_Additional_Examples), /content/body/p/span, line 1, column 23
\)





















Three state system

Consider a spin-1 particle where σ=1,0,+1. We model this with the single particle Hamiltonian

ˆh=μ0Hσ+Δ(1σ2) .

We can also interpret this as describing a spin if σ=±1 and a vacancy if σ=0. The parameter Δ then represents the vacancy formation energy. The single particle partition function is

ζ=Treβˆh=eβΔ+2cosh(βμ0H) .

With NS distinguishable noninteracting spins ( at different sites in a crystalline lattice), we have \boldsymbol{Z=\zeta^\NS} and

FNSf=kBTlnZ=NSkBTln[eβΔ+2cosh(βμ0H)] ,

where f=kBTlnζ is the free energy of a single particle. Note that

ˆnV=1σ2=ˆhΔNNˆm=μ0σ=ˆhH

are the vacancy number and magnetization, respectively. Thus,

nV=ˆnV=fΔ=eΔ/kBTeΔ/kBT+2cosh(μ0H/kBT)

and

m=ˆm=fH=2μ0sinh(μ0H/kBT)eΔ/kBT+2cosh(μ0H/kBT) .

At weak fields we can compute

χT=mH|H=0=μ20kBT22+eΔ/kBT .

We thus obtain a modified Curie law. At temperatures TΔ/kB, the vacancies are frozen out and we recover the usual Curie behavior. At high temperatures, where TΔ/kB, the low temperature result is reduced by a factor of 23, which accounts for the fact that one third of the time the particle is in a nonmagnetic state with σ=0.

Spins and vacancies on a surface

A collection of spin-12 particles is confined to a surface with N sites. For each site, let σ=0 if there is a vacancy, σ=+1 if there is particle present with spin up, and σ=1 if there is a particle present with spin down. The particles are non-interacting, and the energy for each site is given by ε=Wσ2, where W<0 is the binding energy.

  • Let Q=N+N be the number of spins, and N0 be the number of vacancies. The surface magnetization is M=NN. Compute, in the microcanonical ensemble, the statistical entropy S(Q,M).
  • Let q=Q/N and m=M/N be the dimensionless particle density and magnetization density, respectively. Assuming that we are in the thermodynamic limit, where N, Q, and M all tend to infinity, but with q and m finite, Find the temperature T(q,m). Recall Stirling’s formula

    ln(N!)=NlnNN+O(lnN) .

  • Show explicitly that T can be negative for this system. What does negative T mean? What physical degrees of freedom have been left out that would avoid this strange property?

There is a constraint on N, N0, and N:

N+N0+N=Q+N0=N .

The total energy of the system is E=WQ.

  • The number of states available to the system is

    Ω=N!N!N0!N! .

    Fixing Q and M, along with the above constraint, is enough to completely determine {N,N0,N}:

    N=12(Q+M),N0=NQ,N=12(QM) ,

    whence

    Ω(Q,M)=N![12(Q+M)]![12(QM)]!(NQ)! .

    The statistical entropy is S=kBlnΩ:

    S(Q,M)=kBln(N!)kBln[12(Q+M)!]kBln[12(QM)!]kBln[(NQ)!] .

  • Now we invoke Stirling’s rule,

    ln(N!)=NlnNN+O(lnN) ,

    to obtain

    lnΩ(Q,M)=NlnNN12(Q+M)ln[12(Q+M)]+12(Q+M)12(QM)ln[12(QM)]+12(QM)NN(NQ)ln(NQ)+(NQ)=NlnN12Qln[14(Q2M2)]12Mln(Q+MQM)

    Combining terms,

    lnΩ(Q,M)=Nqln[12q2m2]12Nmln(q+mqm)N(1q)ln(1q) ,

    where Q=Nq and M=Nm. Note that the entropy S=kBlnΩ is extensive. The statistical entropy per site is thus

    s(q,m)=kBqln[12q2m2]12kBmln(q+mqm)kB(1q)ln(1q) .

    The temperature is obtained from the relation

    1T=(SE)M=1W(sq)m=NN1Wln(1q)1Wln[12q2m2] .

    Thus,

    T=W/kBln[2(1q)/q2m2] .

  • We have 0q1 and qmq, so T is real (thank heavens!). But it is easy to choose {q,m} such that T<0. For example, when m=0 we have T=W/kBln(2q12) and T<0 for all q(23,1]. The reason for this strange state of affairs is that the entropy S is bounded, and is not an monotonically increasing function of the energy E (or the dimensionless quantity Q). The entropy is maximized for N↑=N0=N=13, which says m=0 and q=23. Increasing q beyond this point (with m=0 fixed) starts to reduce the entropy, and hence (S/E)<0 in this range, which immediately gives T<0. What we’ve left out are kinetic degrees of freedom, such as vibrations and rotations, whose energies are unbounded, and which result in an increasing S(E) function.

Fluctuating Interface

Consider an interface between two dissimilar fluids. In equilibrium, in a uniform gravitational field, the denser fluid is on the bottom. Let z=z(x,y) be the height the interface between the fluids, relative to equilibrium. The potential energy is a sum of gravitational and surface tension terms, with

Ugrav=d2xz0dzΔρgzUsurf=d2x12σ(z)2 .

We won’t need the kinetic energy in our calculations, but we can include it just for completeness. It isn’t so clear how to model it a priori so we will assume a rather general form

T=d2xd2x12μ(x,x)z(x,t)tz(x,t)t .

We assume that the (x,y) plane is a rectangle of dimensions Lx×Ly. We also assume μ(x,x)=μ(|xx|). We can then Fourier transform

z(x)=(LxLy)1/2kzkeikx ,

where the wavevectors k are quantized according to

k=2πnxLxˆx+2πnyLyˆy ,

with integer nx and ny, if we impose periodic boundary conditions (for calculational convenience). The Lagrangian is then

L=12k[μk|˙zk|2(gΔρ+σk2)|zk|2] ,

where

μk=d2xμ(|x|)eikx .

Since z(x,t) is real, we have the relation zk=zk, therefore the Fourier coefficients at k and k are not independent. The canonical momenta are given by

pk=L˙zk=μk˙zk,pk=L˙zk=μk˙zk

The Hamiltonian is then

ˆH=k[pkzk+pkzk]L=k[|pk|2μk+(gΔρ+σk2)|zk|2] ,

where the prime on the k sum indicates that only one of the pair {k,k} is to be included, for each k.

We may now compute the ordinary canonical partition function:

Z=kd2pkd2zk(2π)2e|pk|2/μkkBTe(gΔρ+σk2)|zk|2/kBT=k(kBT2)2(μkgΔρ+σk2) .

Thus,

F=kBTkln(kBT2Ωk) ,

where13

Ωk=(gΔρ+σk2μk)1/2 .

is the normal mode frequency for surface oscillations at wavevector k. For deep water waves, it is appropriate to take μk=Δρ/|k|, where Δρ=ρLρGρL is the difference between the densities of water and air.

It is now easy to compute the thermal average

|zk|2=d2zk|zk|2e(gΔρ+σk2)|zk|2/kBT/d2zke(gΔρ+σk2)|zk|2/kBT=kBTgΔρ+σk2 .

Note that this result does not depend on μk, on our choice of kinetic energy. One defines the correlation function

C(x)z(x)z(0)=1LxLyk|zk|2eikx=d2k(2π)2(kBTgΔρ+σk2)eikx=kBT4πσ0dqeik|x|q2+ξ2=kBT4πσK0(|x|/ξ) ,

where ξ=gΔρ/σ is the correlation length, and where K0(z) is the Bessel function of imaginary argument. The asymptotic behavior of K0(z) for small z is K0(z)ln(2/z), whereas for large z one has K0(z)(π/2z)1/2ez. We see that on large length scales the correlations decay exponentially, but on small length scales they diverge. This divergence is due to the improper energetics we have assigned to short wavelength fluctuations of the interface. Roughly, it can cured by imposing a cutoff on the integral, or by insisting that the shortest distance scale is a molecular diameter.

Dissociation of Molecular Hydrogen

Consider the reaction

H\ooalign{\raise1pt\hbox{\relbar\joinrel\joinrel}\crcr  \lower1pt\hbox{\joinrel\relbar\joinrel}}p++e.

In equilibrium, we have

μH=μp+μe .

What is the relationship between the temperature T and the fraction x of hydrogen which is dissociated?

Let us assume a fraction x of the hydrogen is dissociated. Then the densities of H, p, and e are then

nH=(1x)n,np=xn,ne=xn .

The single particle partition function for each species is

ζ=gNN!(Vλ3T)NeNεint/kBT ,

where g is the degeneracy and εint the internal energy for a given species. We have εint=0 for p and e, and εint=Δ for H, where Δ=e2/2aB=13.6eV, the binding energy of hydrogen. Neglecting hyperfine splittings14, we have gH=4, while ge=gp=2 because each has spin S=12. Thus, the associated grand potentials are

ΩH=gHVkBTλ3T,He(μH+Δ)/kBTΩp=gpVkBTλ3T,peμp/kBTNNΩe=geVkBTλ3T,eeμe/kBT ,

where

λT,a=2π2makBT

for species a. The corresponding number densities are

n=1V(Ωμ)T,V=gλ3Te(μεint)/kBT ,

and the fugacity z=eμ/kBT of a given species is given by

z=g1nλ3Teεint/kBT .

We now invoke μH=μp+μe, which says zH=zpze, or

g1HnHλ3T,HeΔ/kBT=(g1pnpλ3T,p)(g1eneλ3T,e) ,

which yields

(x21x)n˜λ3T=eΔ/kBT ,

where ˜λT=2π2/mkBT, with m=mpme/mHme. Note that

˜λT=aB4πmHmpΔkBT ,

where aB=0.529Å is the Bohr radius. Thus, we have

(x21x)(4π)3/2ν=(TT0)3/2eT0/T ,where T0=Δ/kB=1.578×105K and ν=na3B. Consider for example a temperature T=3000K, for which T0/T=52.6, and assume that x=12. We then find ν=1.69×1027, corresponding to a density of n=1.14×102cm3. At this temperature, the fraction of hydrogen molecules in their first excited (2s) state is xeT0/2T=3.8×1012. This is quite striking: half the hydrogen atoms are completely dissociated, which requires an energy of Δ, yet the number in their first excited state, requiring energy 12Δ, is twelve orders of magnitude smaller. The student should reflect on why this can be the case.


This page titled 4.10: Appendix I- Additional Examples is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

Support Center

How can we help?