G5.651: Statistical Mechanics Notes for Lecture 5 From the classical virial theorem I. TEMPERATURE AND PRESSURE ESTIMATORS hx i x j i = kt ij we arrived at the equipartition theorem: * + p i = m i NkT where p 1 ; :::; p N are the N Cartesian momenta of the N particles in a system. This says that the microscopic function of the N momenta that corresponds to the temperature, a macroscopic observable of the system, is given by K(p 1 ; :::; p N ) = p i m i The ensemble average of K can be related directly to the temperature T = Nk hk(p 1; :::; p N )i = nr hk(p 1; :::; p )i K(p 1 ; :::; p N ) is known as an estimator (a term taken over from the Monte Carlo literature) for the temperature. An estimator is some function of the phase space coordinates, i.e., a function of microscopic states, whose ensemble average gives rise to a physical observable. An estimator for the pressure can be derived as well, starting from the basic thermodynamic relation: A ln Q(N; V; T ) P =? = kt V V with Q(N; V; T ) = C N dx e?(x) = C N d N p V d N r e?(p;r) The volume dependence of the partition function is contained in the limits of integration, since the range of integration for the coordinates is determined by the size of the physical container. For example, if the system is conned within a cubic box of volume V = L, with L the length of a side, then the range of each q integration will be from to L. If a change of variables is made to s i = q i =L, then the range of each s integration will be from to 1. The coordinates s i are known as scaled coordinates. For containers of a more general shape, a more general transformation is s i = V?1= r i In order to preserve the phase space volume element, however, we need to ensure that the transformation is a canonical one. Thus, the corresponding momentum transformation is i = V 1= p i With this coordinate/momentum transformation, the phase space volume element transforms as d N p d N r = d N d N s 1
Thus, the volume element remains the same as required. With this transformation, the amiltonian becomes = p i m i + U(r 1 ; :::; r N ) = and the canonical partition function becomes Q(N; V; T ) = C N d N d N s exp (? V?= i m i + U(V 1= s 1 ; :::; V 1= s N ) " V?= i m i + U(V 1= s 1 ; :::; V 1= s N ) Thus, the pressure can now be calculated by explicit dierentiation with respect to the volume, V : P = kt 1 Q Q V = kt Q C N = kt Q C N = 1 V = h? V i Thus, the pressure estimator is and the pressure is given by d N d N p d N s d N r " "?5= V V * p i m i + r i F i + p i? m i V (p 1 ; :::; p N ; r 1 ; :::; r N ) = (x) = 1 V P = h(x)i i m i? V?= # r i U r i s i e?(p;r) p i m i + r i F i (r) #) # U e? (V 1= s i ) For periodic systems, such as solids and currently used models of liquids, an absolute Cartesian coordinate q i is ill-dened. Thus, the virial part of the pressure estimator P i q if i must be rewritten in a form appropriate for periodic systems. This can be done by recognizing that the force F i is obtained as a sum of contributions F ij, which is the force on particle i due to particle j. Then, the classical virial becomes r i F i = = 1 = 1 = 1 r i X j6=i F ij 4 X i 4 X i X i;j;i6=j r i X j6=i r i X j6=i F ij + X j F ij? X j (r i? r j ) F ij = 1 r j X i6=j r j X i6=j X i;j;i6=j F ji 5 F ij 5 r ij F ij where r ij is now a relative coordinate. r ij must be computed consistent with periodic boundary conditions, i.e., the relative coordinate is dened with respect to the closest periodic image of particle j with respect to particle i. This gives rise to surface contributions, which lead to a nonzero pressure, as expected.
II. ENERGY FLUCTUATIONS IN TE CANONICAL ENSEMBLE In the canonical ensemble, the total energy is not conserved. ((x) 6= const). What are the uctuations in the energy? The energy uctuations are given by the root mean square deviation of the amiltonian from its average hi: E = q h(? hi) i = p h i? hi Therefore But Thus, hi =? h i = 1 Q C N = 1 Q C N ln Q(N; V; T ) = 1 Q Q dx (x)e?(x) dx e?(x) = ln Q + 1 Q Q = ln Q + 1 Q Q = ln Q + ln Q h i? hi = ln Q ln Q = kt C V E = p kt C V Therefore, the relative energy uctuation E=E is given by p E kt E = C V E Now consider what happens when the system is taken to be very large. In fact, we will dene a formal limit called the thermodynamic limit, in which N?! 1 and V?! 1 such that N=V remains constant. Since C V and E are both extensive variables, C V N and E N, E E 1 p N?! as N! 1 But E=E would be exactly in the microcanonical ensemble. Thus, in the thermodynamic limit, the canonical and microcanonical ensembles are equivalent, since the energy uctuations become vanishingly small.
III. ISOTERMAL-ISOBARIC ENSEMBLE A. Basic Thermodynamics The elmholtz free energy A(N; V; T ) is a natural function of N, V and T. The isothermal-isobaric ensemble is generated by transforming the volume V in favor of the pressure P so that the natural variables are N, P and T (which are conditions under which many experiments are performed { \standard temperature and pressure," for example). Performing a Legendre transformation of the elmholtz free energy But Thus, ~A(N; P; T ) = A(N; V (P ); T )? V (P ) A A V =?P V where G(N; P; T ) is the Gibbs free energy. ~A(N; P; T ) = A(N; V (P ); T ) + P V G(N; P; T ) The dierential of G is G dg = P G dp + T G dt + N P;T dn But from G = A + P V, we have dg = da + P dv + V dp but da =?SdT? P dv + dn, thus dg =?SdT + V dp + dn Equating the two expressions for dg, we see that G V = P G S =? T G = N P;T B. The partition function and relation to thermodynamics In principle, we should derive the isothermal-isobaric partition function by coupling our system to an innite thermal reservoir as was done for the canonical ensemble and also subject the system to the action of a movable piston under the inuence of an external pressure P. In this case, both the temperature of the system and its pressure will be controlled, and the energy and volume will uctuate accordingly. owever, we saw that the transformation from E to T between the microcanonical and canonical ensembles turned into a Laplace transform relation between the partition functions. The same result holds for the transformation from V to T. The relevant \energy" quantity to transform is the work done by the system against the external pressure P in changing its volume from V = to V, which will be P V. Thus, the isothermal-isobaric partition function can be expressed in terms of the canonical partition function by the Laplace transform: 4
(N; P; T ) = 1 V 1 where V is a constant that has units of volume. Thus, 1 1 (N; P; T ) = V N!h N dv The Gibbs free energy is related to the partition function by dv e?p V Q(N; V; T ) G(N; P; T ) =? 1 ln (N; P; T ) dxe?((x)+p V ) This can be shown in a manner similar to that used to prove the A =?(1=) ln Q. The dierential equation to start with is Other thermodynamic relations follow: Volume: G = A + P V = A + P G P V =?kt ln (N; P; T ) P Enthalpy: = h(x) + P V i =? ln (N; P; T ) eat capacity at constant pressure C P = T = k ln (N; P; T ) Entropy: G S =? T = k ln (N; P; T ) + T The uctuations in the enthalpy are given, in analogy with the canonical ensemble, by so that = pkt C P = p kt C P so that, since C P and are both extensive, = 1= p N which vanish in the thermodynamic limit. 5
C. Pressure and work virial theorems As noted earlier, the quantity?=v is a measure of the instantaneous value of the internal pressure P int. Let us look at the ensemble average of this quantity 1 hp int i =? 1 C N dv e?p V dx V e?(x) = 1 C N = 1 Doing the volume integration by parts gives Thus, hp int i = 1 = P 1 1 1 e?p V kt Q(N; V; T ) 1 1 dv e?p V dxkt V e?(x) dv e?p V kt Q(N; V; T ) V? 1 dv e?p V Q(N; V; T ) = P 1 hp int i = P dv kt V e?p V Q(N; V; T ) This result is known as the pressure virial theorem. It illustrates that the average of the quantity?=v gives the xed pressure P that denes the ensemble. Another important result comes from considering the ensemble average hp int V i hp int V i = 1 1 e?p V kt V Q(N; V; T ) 1 1 dv e?p V kt V Q(N; V; T ) V Once again, integrating by parts with respect to the volume yields 1 hp int V i = 1? 1 dv kt V V e?p V Q(N; V; T ) or = 1?kT dv e?p V Q(V ) + P =?kt + P hv i 1 hp int V i + kt = P hv i dv e?p V V Q(V ) This result is known as the work virial theorem. It expresses the fact that equipartitioning of energy also applies to the volume degrees of freedom, since the volume is now a uctuating quantity. 6