4.3: Thermal Equilibrium
- Page ID
- 18564
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\( \newcommand\Vom
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\( \newcommand\Vpi
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\( \newcommand\Vvarpi
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\( \newcommand\Vrho
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\( \newcommand\Vvarrho
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\( \newcommand\Vsigma
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\( \newcommand\Vvarsigma
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\( \newcommand\Vtau
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\( \newcommand\Vupsilon
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\( \newcommand\Vphi
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\( \newcommand\Vvarphi
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\( \newcommand\Vchi
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\( \newcommand\Vpsi
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\( \newcommand\Vomega
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\( \newcommand\VGamma
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Two Systems in Thermal Contact
Consider two systems in thermal contact, as depicted in Figure \(\PageIndex{1}\). The two subsystems #1 and #2 are free to exchange energy, but their respective volumes and particle numbers remain fixed. We assume the contact is made over a surface, and that the energy associated with that surface is negligible when compared with the bulk energies \(E\ns_1\) and \(E\ns_2\). Let the total energy be \(E=E\ns_1+E\ns_2\). Then the density of states \(D(E)\) for the combined system is
\[D(E)=\int\!\!dE\ns_1\,D\ns_1(E\ns_1)\,D\ns_2(E-E\ns_1)\ .\]
The probability density for system #1 to have energy \(E\ns_1\) is then
\[P\ns_1(E\ns_1)={D\ns_1(E\ns_1)\,D\ns_2(E-E\ns_1)\over D(E)}\ .\]
Note that \(P\ns_1(E\ns_1)\) is normalized: \(\int\!dE\ns_1 \,P\ns_1(E\ns_1)=1\). We now ask: what is the most probable value of \(E\ns_1\)? We find out by differentiating \(P\ns_1(E\ns_1)\) with respect to \(E\ns_1\) and setting the result to zero. This requires
\[\begin{split} 0&={1\over P\ns_1(E\ns_1)}\,{dP\ns_1(E\ns_1)\over dE\ns_1} = {\pz\over \pz E\ns_1}\,\ln P\ns_1(E\ns_1)\\ &={\pz\over \pz E\ns_1}\,\ln D\ns_1(E\ns_1) + {\pz\over \pz E\ns_1}\,\ln D\ns_2(E-E\ns_1)\ . \label{maxprob} \end{split}\]
We conclude that the maximally likely partition of energy between systems #1 and #2 is realized when
\[{\pz S\ns_1\over \pz E\ns_1}={\pz S\ns_2\over \pz E\ns_2}\ .\]
This guarantees that
\[S(E,E\ns_1)=S\ns_1(E\ns_1) + S\ns_2(E-E\ns_1)\]
is a maximum with respect to the energy \(E\ns_1\), at fixed total energy \(E\).
The temperature \(T\) is defined as
\[{1\over T}=\pabc{S}{E}{V,N}\ , \label{Teqn}\]
a result familiar from thermodynamics. The difference is now we have a more rigorous definition of the entropy. When the total entropy \(S\) is maximized, we have that \(T\ns_1=T\ns_2\). Once again, two systems in thermal contact and can exchange energy will in equilibrium have equal temperatures.
According to Equations \ref{phinrel} and \ref{phiurel}, the entropies of nonrelativistic and ultrarelativistic ideal gases in \(d\) space dimensions are given by
\[\begin{align} S\ns{_{\ssr{NR}}}&=\half Nd\,\kB\ln\!\bigg({E\over N}\bigg) + N\kB\ln\!\bigg({V\over N}\bigg) + { const.}\\ S\ns_{\ssr{UR}}&=Nd\,\kB\ln\!\bigg({E\over N}\bigg) + N\kB\ln\!\bigg({V\over N}\bigg) + { const.}\ .\end{align}\]
Invoking Equation \ref{Teqn}, we then have
\[E\ns_{\ssr{NR}}=\half N d\,\kT \qquad,\qquad E\ns_{\ssr{UR}}=N d\,\kT\ .\]
We saw that the probability distribution \(P\ns_1(E\ns_1)\) is maximized when \(T\ns_1=T\ns_2\), but how sharp is the peak in the distribution? Let us write \(E\ns_1=E^*_1+\RDelta E\ns_1\), where \(E^*_1\) is the solution to Equation \ref{maxprob}. We then have
\[\ln P\ns_1(E^*_1+\RDelta E\ns_1)=\ln P\ns_1(E^*_1) + {1\over 2\kB}\, {\pz^2\!S\ns_1\over\pz E_1^2}\bigg|\nd_{E^*_1}(\RDelta E\ns_1)^2 + {1\over 2\kB}\,{\pz^2\!S\ns_2\over\pz E_2^2}\bigg|\nd_{E^*_2} (\RDelta E\ns_1)^2+ \ldots\ ,\]
where \(E^*_2=E-E^*_1\). We must now evaluate
\[{\pz^2 \!S\over \pz E^2}={\pz\over\pz E}\bigg({1\over T}\bigg)=-{1\over T^2}\pabc{T}{E}{V,N} =-{1\over T^2 \,C\ns_V}\ ,\]
where \(C\ns_V=\big(\pz E/\pz T\big)\nd_{V,N}\) is the heat capacity. Thus,
\[P\ns_1 = P^*_1\,e^{-(\RDelta E\ns_1)^2/2k\ns_\RB T^2 {\bar C}\ns_V}\ ,\]
where
\[{\bar C}\ns_V={C\ns_{V,1}\,C\ns_{V,2}\over C\ns_{V,1}+C\ns_{V,2}}\ .\]
The distribution is therefore a Gaussian, and the fluctuations in \(\RDelta E\ns_1\) can now be computed:
\[\big\langle (\RDelta E\ns_1)^2\big\rangle = \kB T^2\,{\bar C}\ns_V \qquad\Longrightarrow\qquad (\RDelta E_1)\ns_{\ssr{RMS}}=\kT\sqrt{\bar C \ns_V/\kB}\ .\]
The individual heat capacities \(C\ns_{V,1}\) and \(C\ns_{V,2}\) scale with the volumes \(V\ns_1\) and \(V\ns_2\), respectively. If \(V\ns_2\gg V\ns_1\), then \(C\ns_{V,2}\gg C\ns_{V,1}\), in which case \({\bar C}\ns_V\approx C\ns_{V,1}\). Therefore the RMS fluctuations in \(\RDelta E\ns_1\) are proportional to the square root of the system size, whereas \(E\ns_1\) itself is extensive. Thus, the ratio \((\RDelta E_1)\ns_{\ssr{RMS}}/E\ns_1\propto V^{-1/2}\) scales as the inverse square root of the volume. The distribution \(P\ns_1(E\ns_1)\) is thus extremely sharp.
Thermal, mechanical and chemical equilibrium
We have \(dS\big|\nd_{V,N}={1\over T}\,dE\) , but in general \(S=S(E,V,N)\). Equivalently, we may write \(E=E(S,V,N)\). The full differential of \(E(S,V,N)\) is then \(dE=T\,dS - p\,dV + \mu\,dN\), with \(T=\pabc{E}{S}{\,V,N}\) and \(p=-\pabc{E}{V}{\,S,N}\) and \(\mu=\pabc{E}{N}{\,S,V}\). As we shall discuss in more detail, \(p\) is the pressure and \(\mu\) is the chemical potential. We may thus write the total differential \(dS\) as
\[dS={1\over T}\,dE+ {p\over T}\,dV - {\mu\over T}\,dN\quad.\]
Employing the same reasoning as in the previous section, we conclude that entropy maximization for two systems in contact requires the following:
- If two systems can exchange energy, then \(T\ns_1=T\ns_2\). This is thermal equilibrium.
- If two systems can exchange volume, then \(p\ns_1/T\ns_1=p\ns_2/T\ns_2\). This is mechanical equilibrium.
- If two systems can exchange particle number, then \(\mu\ns_1/T\ns_1=\mu\ns_2/T\ns_2\). This is chemical equilibrium.
Gibbs-Duhem Relation
The energy \(E(S,V,N)\) is an extensive function of extensive variables, it is homogeneous of degree one in its arguments. Therefore \(E(\lambda S,\lambda V,\lambda N)=\lambda E\), and taking the derivative with respect to \(\lambda\) yields
\[\begin{split} E&=S\pabc{E}{S}{V,N}+ V\pabc{E}{V}{S,N}+N\pabc{E}{N}{S,V}\\ &=TS-pV+\mu N\quad. \end{split}\]
Taking the differential of each side, using the Leibniz rule on the RHS, and plugging in \(dE=T\,dS-p\,dV+\mu\,dN\), we arrive at the Gibbs-Duhem relation5,
\[S\,dT-V dp + N\,d\mu=0\quad.\]
This, in turn, says that any one of the intensive quantities \((T,p,\mu)\) can be written as a function of the other two, in the case of a single component system.