After a historical introduction, this book presents chapters about thermodynamics, ensemble theory, simple gases theory, Ideal Bose and Fermi systems, statistical mechanics of interacting systems, phase transitions, and computer simulations. This edition includes new topics such as Bose-Einstein condensation and degenerate Fermi gas behavior in ultracold atomic gases and chemical equilibrium. It also explains the correlation functions and scattering; fluctuation-dissipation theorem and the dynamical structure factor; phase equilibrium and the Clausius-Clapeyron equation; and exact solutions of one-dimensional fluid models and two-dimensional Ising model on a finite lattice.
New topics can be found in the appendices, including finite-size scaling behavior of Bose-Einstein condensates, a summary of thermodynamic assemblies and associated statistical ensembles, and pseudorandom number generators. Other chapters are dedicated to two new topics, the thermodynamics of the early universe and the Monte Carlo and molecular dynamics simulations.
This book is invaluable to students and practitioners interested in statistical mechanics and physics. Pathria Statistical Mechanics Solution Pdf. The liquid side of solid—liquid lines will start at Pt and extend upward as in figure 6. Chapter 5 5. For a formal solution to this problem, see Kubo , problem 2. If we use the unsymmetrized wave function 5. The diagonal elements of the density matrix then are h1,. The structure of expressions 1 and 2 shows that there is no spatial correlation among the particles of this system.
For the second part, we substitute 2 into 1 and integrate over the posi- tion coordinates of the particles. For solutions to these problems, consult the references cited in Notes 10 and Chapter 6 6. In the B. Starting with eqn. It is then straightforward to see, with the help of the formulae B. For light emitted in the x-direction, only the x-component of the molecular velocity u will contribute to the Doppler effect. The extra contribution comes from the potential energy of the system, which also rises with T.
Note, from eqns. Drawing the uranium hexafluoride gas from near the center of the cylinder results in a sample that is isotopically enhanced with U compared to the input concentration. This process may be repeated as often as needed to achieve the isotopic fraction needed. Equating this result with 1, we obtain the desired expression for C. Ordinarily, when a molecule is reflected from a stationary wall that is perpendicular to the z-direction, the z-component of its velocity u simply changes sign, i.
We refer to expression 6. Proceeding as Section 6. It is not difficult to show, see the corresponding calculation in Problem 6. As for explicit variations with t, we make use of eqn. The variations of N and P with t follow straightforwardly. The latter result implies that vr. We extend the treatment of Problem 3. The contribution from mode 6 is about 0.
The net result is: 3. Equation 6. Using equation 6. We now write eqn. This leads to the desired result 7. The desired result now follows readily. In view of the fact that S, at constant N, is a function of z only, see eqn. The other result follows straight- forwardly. Multiplying the two, we obtain the desired result. An explicit ex- pression for this quantity can be written down using the result quoted in Problem 7. The critical behavior of these quantities is also straightforward to check.
Using a result from Problem 7. Under the conditions of this problem, the summation in eqn. Expression 7. The relative mean-square fluctuation in N is given by the general for- mula 4. The mean-square fluctuation in E is given by the general formula 4. The second term can be evaluated with the help of eqns. Thus, all in all, the relative fluctuation in E is negligible at all T. More accurately, the phenomenon of condensation requires that both Ne and N0 be of order N. A glance at eqn. To study the specific heats we first observe, from eqns.
At long-time, the width of the distribution grows linearly in time. Integrating equation 7. Using expressions 7. Just as in Problem 1.
The number density of photons in the cosmic microwave background CMB follows from equation 7. According to Sec. Using the Debye spectrum 7. Hence the stated result. The specific heat of the system is given by the general expression 7. The mode density in this case is given by, see eqn. The rest of the argument is similar to the one made in the previous prob- lem; the net result is that the specific heat of the given system, at low temperatures, is proportional to T n.
The Hamiltonian of this system is given by eqn. In the present case, eqn. Following Secs. We write eqn. Chapter 8 8. Referring to Fig. The reason for the numerical discrepancy lies in the fact that the present approximation takes into account only a fraction of the particles that are thermally excited; see Fig.
This problem is similar to Problem 7. To obtain the various low-temperature expressions, we make use of expansions 8. At low temperatures, using formula E.
Problem 6. Taking all the nucleons together, this gives a particle density of about 8. Substituting this into eqn. Equation 2 then gives the desired result 8.
Next, we have from eqns. Parts i and ii are straightforward. Eliminating T among these relations, we obtain the desired equation of an adiabat. For part v , we proceed as follows. See also Problem 8. In the notation of Sec. We observe that eqn. Utilizing the result obtained in Problem 8. Substituting this result into 1 and making use of eqn. Using the Friedmann equation 9.
Just use equations 9. This is the justification for treating the relativistic electrons and positrons as noninteracting. Equation 9. After the density of electrons levels off at the nearly the proton density, you can use equation 9. Then the positron number density is given by equation 9.
After the electron—positron annihilation, the only relativistic species left are the photons and the neutrinos. Following the solution to problem 9. If the current CMB temperature was 27K rather than 2. If the current CMB temperature were 0.
The strong interaction exhibits asymptotic freedom at high energies jus- tifying treating the quarks an gluons as noninteracting. The effective number of species in equilibrium in these tiny quark—gluon plasmas is accounted for using only the up and down quarks and the gluons. Pho- tons, and leptons, for example, easily escape without interacting with the plasma. This is the record hottest temperature for matter created in the laboratory.
Proceeding as in problem 9. Chapter 10 For the rest of the question, follow the solution to Problem For this problem, we integrate With the given interparticle interaction, eqn. We consider a volume element dx 1 dy 1 dz 1 around the point P x1 , 0, 0 in solid 1 and a volume element dx 2 dy 2 dz 2 around the point Q x2 , y2 , z2 in solid 2.
Substituting these results into eqn. Comparing this with eqn. This will lead to the quoted expression for the coefficient bs2.
This will lead to the quoted expression for the coefficient bA 2. Next, using eqns. The coefficients of those terms are integrals over the Mayer functions that are continuous functions of r12 even for the infinite step function potential; see equation For a complete solution to these problems, see Landau and Lifshitz , sec.
Chapter 11 For a solution to this problem, see Feynman For solutions to these problems, see Fetter , Comparing this result with eqn. For a complete solution to this problem, see the first edition of this book — Sec. Use the dimensionless form from problem Equation Chapter 12 The critical amplitudes as well as the critical constants Pc , vc and Tc , however, do vary with n and hence are model-dependent.
For the critical constants of the gas, we first note from eqn. Let us concentrate on one particular spin, s0 , in the lattice and look at the Pq part of the energy E that involves this spin, viz. However, this is only a high-temperature approximation, so no firm conclusion about a phase transition can be drawn from it. For that, we must look into the possibility of spontaneous magnetization in the system.
We shall consider only the Heisenberg model; the study of the Ising model is somewhat simpler. Following the procedure of Problem In analogy with eqn. For a complete solution to this problem, see Kubo , problem 5.
P b As in Problem The singularity in question is, therefore, precisely the same as the one encountered in Sec. Using eqn. We refer to the solutions to Problems Now, substituting these values of r1 and s0 into eqns.
We must, in this context, remember that only a positive A will yield a real m0. The foregoing observations should suffice to prove statements a , e , f and g of this problem. It now follows from eqn. These observations should suffice to prove statements b , c and d.
Substituting Similarly, substituting By the scaling hypothesis of Sec. Now, in view of eqn. Chapter 13 The partition function In the notation of Problem To compare these results with the ones following from the Bethe approx- imation, we first observe that eqn.
As for eqn. Hence the equivalence of the two treatments. Now, a reference to eqns. The second term in the Hamiltonian will contribute towards the short-range order in the system.
The order-disorder transition is made possible by the fact that the resulting interaction is of an infinite range. The study of the various thermodynamic properties of the system is now straightfor- ward.
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