Plischke bergersen pdf

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Privacy Overview. Magnanti , James B. Steven C. I 6th Ed. II 6th Ed. Undeland, William P. Grainger William D. Viterbi and Jim K. Neftci, B. Yates , David J. Rehg, Glenn J. Roberts, M. Shampine, I. McCormac and Stephen F. Scott MacKenzie and Raphael C. Askeland, Pradeep P. Young, Roger A. In the laboratory, pressures in the range 10 to kbar are fairly standard and Pv, then becomes comparable with the vacancy formation energy.

Show that any phase space trajectory x t with energy E will, on the average, spend equal time in all regions of the constant energy surface I' E. Show that there are regions of the constant energy surface which are not visited for any particular trajectory z t. If you are not familiar with normal coordinates, a good place to look is Goldstein [94]. Gibbs Paradoz.

Statistical Ensembles c Estimate the entropy of mixing of 1 mol of Ar and 1 mol of Kr. Consider a system of particles in which the force between the parti- cles is derivable from a potential which is a generalized homogeneous function of degree 7, that is, b U Ary, Ara, Dielectric Function of a Diatomic Gas. The local field corrections will be different for permanent and induced dipoles ].

Equivalence of Thermodynamic and Statistical Mechanical Definitions. Find an estimate for the correction term in analogy with equation 2. Statistical Ensembles 2. Black-body Radiation. In a dilute gas the density will be low enough that the translational motion can be treated classically. Chapter 2. Construct the grand canonical ensemble from the maximum entropy principle by maximizing the entropy, subject to the constraints that the mean energy and particle number are fixed. Effect of Lattice Vibrations on Vacancy Formation.

To establish the effect qualitatively, consider the following crude model. Each atom vibrates as an independent three-dimensional Einstein oscillator of frequency wo. Assume further that if a nearest-neighbor site is vacant, the frequency of the mode corresponding to vibration in the direction of the vacancy changes from wo to w. Let q be the number of nearest neighbors. Each mode then corresponds to the vibration of two springs.

Partition Function at Fixed Pressure. Consider a system of N noninteracting molecules in a container of cross-sectional area A. The top consists of an airtight piston of mass M which slides without friction.

You may neglect the effect of gravity on the gas molecules. Stability of a white dwarf against gravitational collapse. It is energetically favorable for a body held together by gravita- tional forces to be as compact as possible.

We take the star to be made up of an approximately equal number N of electrons and protons, since otherwise the Coulomb repulsion would overcome the gravitational in- teraction. Somewhat arbitrarily we also assume that there is an equal number of neutrons and protons. On earth the gravitational pressure is not large enough to overcome the repulsive forces between atoms and molecules at short distance. Inside the sun, matter does not exist in the form of atoms and molecules, but since it is still burning there is radi- ation pressure which keeps it from collapsing.

Let us consider a burnt out star such as a white dwarf. Because of their large mass the kinetic energy of the protons and neutrons will be small compared to that of the electrons. Let my be the nucleon mass. Assume the mass density is approximately constant inside the star. Show that, if there is an equal number of protons and neutrons, the potential energy will be given by 12 N? Find the radius for which the potential energy plus kinetic energy is a minimum for a white dwarf with the same mass as the sun 1.

Since for large N we have N? The star will then collapse. Use the relativistic formula 2. In recent years a great deal of progress has been achieved in our understanding of phase transitions, notably through the development of the renormalization group approach of Wilson, Fisher, Kadanoff, and others.

We postpone a discussion of this theory until Chapters 5 and 6. In this chapter we discuss an older approach known as mean field theory, which generally gives us a qualitative description of the phenomena of interest. A common feature of mean field theories is the identification of an order parameter. One approach is to express an approximate free energy in terms of this parameter and minimize the free energy with respect to the order parameter we have used this approach in Section 2.

Another, often equivalent approach is to approximate an interacting system by a noninteracting system in a self-consistent external field expressed in terms of the order parameter. To understand the phenomena associated with the sudden changes in the material properties which take place during a phase transition, it has proven most useful to work with simplified models that single out the essential aspects of the problem.

One important such model, the Ising model, is introduced in Section 3. In Section 3. Mean Field and Landau Theory discuss the same model in the Bragg-Williams approximation, which is a free energy minimization approach.

Further applications are given as problems. This method gives better numerical values for the crit- ical temperature and other properties of the system. However, we show in Section 3.

The most serious fault of mean field theories lies in the neglect of long- range fluctuations of the order parameter. As we shall see, the importance of this omission depends very much on the dimensionality of the problem, and in problems involving one- and two-dimensional systems the results predicted by mean field theory are often qualitatively wrong.

Because of its close relation to mean field theory we discuss in Section 3. Symmetry considerations are in general important in determining the order of a transition. We illustrate this in Section 3. We conclude our discussion of Landau theory in Sec- tion 3.

Finally, in Section 3. An important reference for much of the material in this chapter is Landau and Lifshitz []. Many examples are discussed in Kubo et al. This exchange interaction has a preferred direction, so that the Ising Hamiltonian is diagonal in the rep- resentation in which each spin S,; is diagonal. The eigenstates of 3. In certain cases the two regimes will be separated by a phase transition, that is, there will be a temperature T.

Mean Field and Landau Theory for all i. We refer to m as the order parameter of the system. Consider the terms in 3. These terms are, with j restricted to nearest neighbor sites of site 0, os.

If we disregard the second term in 3. We obtain the constitutive equation for m. We will show in Section 3. The most general normalized density matrix of the type 3. Physically, the approximation 3. When this approximation does not hold we say that the spins are correlated. These excita- tions disorder the system; the expectation value of the magnetization is zero. A similar argument can be used to give a crude estimate of the transition temperature for the two-dimensional Ising model.

Consider an N x N square lattice with free surfaces. We wish to study the set of excitations of the type shown in the Figure 3. If we start the wall from the left there are at least two, sometimes three choices for the direction of the next step if we neglect the possibility of reaching the upper and lower boundaries. There are N possible starting points for the chain. At high temperatures only the trivial solution exists. As can easily be verified by differentiating 3.

The nature of the phase transition in the alloy system is thus identical to the phase transition of the Ising ferromagnet. This last conclusion is independent of the mean field approximation, as we now show. Consider a crystal structure that can be divided into two sublattices, so that the nearest neighbors to the sites on one of the sublattices belong to the other e.

Thus the two systems have identical thermodynamic properties at all temperatures. In the derivations given above, we allowed the concentrations of the two components of the alloy to vary freely. At other sto- ichiometries the face-centered cubic, more complex cubic structures, or the Ssee e. In general, there is no guar- antee that a particular choice of lattice structure, division into sublattices, or selection of order parameter is the correct one. One should therefore be guided by physical intuition in trying out a number of different alternatives, selecting the one with lowest free energy.

The difference in concentration of each species is analogous to the magnetization in the Ising model, and since we are dealing with a system with fixed magnetization rather than external field , we use the symbol A in preference to G for the free energy.

If A z can be differentiated twice, 3. The resulting free energy lies on the convex envelope of A c. The equilibrium concentrations cy, and cz are given by the lever rule 3. This constitutes the double tangent construction of Figure 3. A simple model for phase separation is found in Problem 3. It is interesting to carry out the equivalent variational treatment of Sec- tion 3. In the binary alloy problem the simplest choice of density matrix is JI: [1] 3.

Mean Field and Landau Theory Figure 3. Material of initial concentration co splits up into two phases with concentrations c; and cz. The amount of material in each phase is given by the double tangent construction.

The completion of this treatment is left as an exercise Problem 3. An extension of the approach, which provides the same results for the order parameter but also yields an expression for the free energy, is due to Fowler and Guggenheim [85].

Consider again the simple Ising model 3. It is possible to improve this approximation in a systematic fashion. We suppose that the lattice has coordination number q and now retain as variables a central spin and its shell of nearest neighbors.

The remainder of the lattice is assumed to act on the nearest-neighbor shell through an effective exchange field which we will calculate self-consistently. The fluctuating field acting on the peripheral spins 7, It is clear that equation 3.

Since 3. The condition for these solutions to exist can be written 3. We see that we have achieved a substantial improvement in the prediction of the critical temperature. This is in agreement with the exact results of Section 3. It is often important to have expressions for the free energy as well as the order parameter. We write the Hamiltonian 3. The free energy associated with 3.

Useful approximations to the free energy can frequently be obtained by substituting approximate expressions into 3. It should be obvious that still better results can be obtained by considering larger clusters. However, all approximations that depend in an essential way on truncation of correlations beyond a certain distance will break down in the vicinity of a critical point.

To show this explicitly, we discuss the critical properties of mean field theories in the next section. We now calculate several other thermodynamic func- tions in the vicinity of the critical point. The determination of the specific heat singularity in the Bethe approxima- tion is somewhat more tedious.

The second term will yield a discontinuity at T. We leave the explicit demonstration of this as an exercise Problem 3. The critical properties of cluster theories thus seem to be in a sense universal and not dependent on the level of sophistication of the approximation.

Another quantity that shows similar behavior in all mean field theories is the critical isotherm m T,,h. In the simplest mean field theory we have 3. Moreover, the result is simple enough that thermodynamic functions can be evaluated without too much difficulty. The solution of 3. This behavior contrasts with that of mean field Section 3. It is interesting to compare in more detail the exact and the mean field solution.

For this reason we plot in Figure 3. In Figure 3. Results from the Bethe approximation are not shown since they agree with the exact ones in this case. In Sections 3. Solid line, exact theory; dotted line, mean field theory. To to kw Figure 3. For h 0 the difference between the exact and approximate calculations is similar to that found for the susceptibility shown in Figure 3. Since tanh BJ 0, and the spin-spin correlation function decays exponentially with increasing j for all nonzero temperatures.

The concept of correlation length will prove most useful later. The divergence of the correlation length at the critical point is a universal feature of continuous phase transitions.

The crucial hypothesis is that in the vicinity of the critical point we may expand the free energy in a power series in the order parameter, which we denote by m.

The equilibrium value of m is then the value that minimizes the free energy. Nevertheless, Landau theory is of great utility as a qualitative tool and also plays an important role, after suitable generalization, in the modern renormalization theory of Wilson. We have tacitly assumed that the field h, which is conjugate to the order parameter m, is zero. We have already encountered this type of expansion in the mean field treatment of the Ising model Section 3. In this case the function G m,T takes the form shown in Figure 3.

We see that the order parameter and specific heat have the same form that we obtained previously in our mean field treatment of the Ising model. We now consider a slightly different situation.

The free energy in this case will be as shown in Figure 3. In this situation a discontinuous jump in the order parameter is expected. To see this, let mo 0 be the location of a minimum of G. We must show that when G mo,T. The case where 6 and c approach zero at the same temperature seems at this stage rather unlikely. Below, we shall illustrate this situation by considering the Maier-Saupe model for nematic liquid crystals, but first consider the general case.

We see, therefore, that the appearance of a cubic term in the Landau expansion signals a first-order phase transition.

A chapter on stochastic processes Luiz de Viveiros statisstical it liked it Sep 21, Do you wish to suggest additional problems? Sara added it Oct 27, Just a moment while we sign you in to your Goodreads account. Principles and Applications Tuck C. Yanist marked it as to-read Sep 13, Oxford University Press The discussion of strongly interacting condensed matter systems has been expanded. Beale Limited preview — No trivia or quizzes yet.

The theories are applied to a number of important systems such as liquids, liquid crystals, polymers, membranes, Bose condensation, superfluidity and superconductivity. The quality and accuracy of these lecture notes and exercises has greatly benefited from statistlcal questions and comments of countless students and correspondents.

The development of the basic tools includes a chapter on computer simulations in which both Monte Carlo method and molecular dynamics are introduced, and a section statistica, Brownian dynamics added. Mathias rated it liked it Feb 23, Do you have a question about the lecture notes [tln, tsl]? Clarendon Press, Oxford Cathy marked it as to-read Dec 21, Tomas Pereira De Vasconcelos equilirium it as to-read Dec 05, The modern theory of phase transitions occupies a central place.