1. For each of the diagrams shown in equation, write down the
corresponding formula in terms of f-functions, and explain why the
symmetry factor gives the correct overall coefficient.Equation:... Get solution

2. Draw all the diagrams, connected or disconnected, representing terms in the configuration integral with four factors of fij. You should find 11 diagrams in total, of which five are connected. Get solution

3. Keeping only the first two diagrams in equation , and approximating N ≈ N − l ≈ N − 2 ≈ ∙ ∙ ∙, expand the exponential in a power series through the third power. Multiply each term out, and show that all the numerical coefficients give precisely the correct symmetry factors for the disconnected diagrams.Equation:... Get solution

4. Draw all the connected diagrams containing four dots. There are six diagrams in total; be careful to avoid drawing two diagrams that look superficially different but are actually the same. Which of the diagrams would remain connected if any single dot were removed? Get solution

5. By changing variables as in the text, express the diagram in equation 8.18 in terms of the same integral as in equation 8.31. Do the same for the last two diagrams in the first line of equation. Which diagrams cannot be written in terms of this basic integral?Equation:... Get solution

6. You can estimate the size of any diagram by realizing that f(r) is of order 1 out to a distance of about the diameter of a molecule, and f ≈ 0 beyond that. Hence, a three-dimensional integral of a product of f’s will generally give a result that is of the order of the volume of a molecule. Estimate the sizes of all the diagrams shown explicitly in equation, and explain why it was necessary to rewrite the series in exponential form. Equation:... Get solution

7. Show that, if you don’t make too many approximations, the exponential series in equation includes the three-dot diagram in equation 8.18. There will be some leftover terms; show that these vanish in the thermodynamic limit.Equation:... Get solution

8. Show that the nth virial coefficient depends on the diagrams in equation that have n dots. Write the third virial coefficient, C(T), in terms of an integral of f-functions. Why it would be difficult to carry out this integral?Equation:... Get solution

9. Show that the Lennard-Jones potential reaches its minimum value at r = r0, and that its value at this minimum is –u0 At what value of r does the potential equal zero? Get solution

10. Use a computer to calculate and plot the second virial coefficient for a gas of molecules interacting via the Lennard-Jones potential, for values of κT/u0 ranging from 1 to 7. On the same graph, plot the data for nitrogen given in Problem, choosing the parameters r0 and u0 so as to obtain a good fit.Problem:Even at low density, real gases don’t quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial expansion,...where the functions B(T), C(T), and so on are called the virial coefficients. When the density of the gas is fairly low, so that the volume per mole is large, each term in the series is much smaller than the one before. In many situations it’s sufficient to omit the third term and concentrate on the second, whose coefficient B(T) is called the second virial coefficient (the first coefficient being 1). Here are some measured values of the second virial coefficient for nitrogen (N2): T (K)B (cm3/mol)100−160200− 35300− 4.24009.050016.960021.3(a) For each temperature in the table, compute the second term in the virial equation, B(T)/(V/n), for nitrogen at atmospheric pressure. Discuss the validity of the ideal gas law under these conditions.(b) Think about the forces between molecules, and explain why we might expect B(T) to be negative at low temperatures but positive at high temperatures.(c) Any proposed relation between P, V, and T , like the ideal gas law or the virial equation, is called an equation of state. Another famous equation of state, which is qualitatively accurate even for dense fluids, is the van der Waals equation,...where a and b are constants that depend on the type of gas. Calculate the second and third virial coefficients (B and C) for a gas obeying the van der Waals equation, in terms of a and b. (Hint: The binomial expansion says that ... provided that |px| ≪ 1. Apply this approximation to the quantity [1 − (nb/V)]−1.)(d) Plot a graph of the van der Waals prediction for B(T), choosing a and b so as to approximately match the data given above for nitrogen. Discuss the accuracy of the van der Waals equation over this range of conditions. (The van der Waals equation is discussed much further in Section 5.3.) Get solution

11. Consider a gas of “hard spheres,” which do not interact at all unless their separation distance is less than r0, in which case then interaction energy is infinite. Sketch the Mayer f-function for this gas, and compute the second virial coefficient. Discuss the result briefly. Get solution

12. Consider a gas of molecules whose interaction energy u(r) is infinite for r r0 and negative for r > r0, with a minimum value of –u0. Suppose further that κT ≫ u0, so you can approximate the Boltzmann factor for r > r0 using ex ≈ 1 + x. Show that under these conditions the second virial coefficient has the form B(T) = b − (a/κT), the same as what you found for a van der Waals gas in Problem. Write the van der Waals constants a and b in terms of r0 and u(r), and discuss the results briefly. Problem:Even at low density, real gases don’t quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial expansion,...where the functions B(T), C(T), and so on are called the virial coefficients. When the density of the gas is fairly low, so that the volume per mole is large, each term in the series is much smaller than the one before. In many situations it’s sufficient to omit the third term and concentrate on the second, whose coefficient B(T) is called the second virial coefficient (the first coefficient being 1). Here are some measured values of the second virial coefficient for nitrogen (N2): T (K)B (cm3/mol)100−160200− 35300− 4.24009.050016.960021.3(a) For each temperature in the table, compute the second term in the virial equation, B(T)/(V/n), for nitrogen at atmospheric pressure. Discuss the validity of the ideal gas law under these conditions.(b) Think about the forces between molecules, and explain why we might expect B(T) to be negative at low temperatures but positive at high temperatures.(c) Any proposed relation between P, V, and T , like the ideal gas law or the virial equation, is called an equation of state. Another famous equation of state, which is qualitatively accurate even for dense fluids, is the van der Waals equation,...where a and b are constants that depend on the type of gas. Calculate the second and third virial coefficients (B and C) for a gas obeying the van der Waals equation, in terms of a and b. (Hint: The binomial expansion says that ... provided that |px| ≪ 1. Apply this approximation to the quantity [1 − (nb/V)]−1.)(d) Plot a graph of the van der Waals prediction for B(T), choosing a and b so as to approximately match the data given above for nitrogen. Discuss the accuracy of the van der Waals equation over this range of conditions. (The van der Waals equation is discussed much further in Section 5.3.) Get solution

13. Use the cluster expansion to write the total energy of a monatomic nonideal gas in terms of a sum of diagrams. Keeping only the first diagram, show that the energy is approximately...Use a computer to evaluate this integral numerically, as a function of T, for the Lennard-Jones potential. Plot the temperature-dependent part of the correction term, and explain the shape of the graph physically. Discuss the correction to the heat capacity at constant volume, and compute this correction numerically for argon at room temperature and atmospheric pressure. Get solution

14. In this section I’ve formulated the cluster expansion for a gas with a fixed number of particles, using the “canonical” formalism of Chapter 6. A somewhat cleaner approach, however, is to use the “grand canonical” formalism introduced in Section 7.1, in which we allow the system to exchange particles with a much larger reservoir.(a) Write down a formula for the grand partition function (Z) of a weakly interacting gas in thermal and diffusive equilibrium with a reservoir at fixed T and µ. Express Z as a sum over all possible particle numbers N, with each term involving the ordinary partition function Z(N).(b) Use equations to express Z(N) as a sum of diagrams, then carry out the sum over N, diagram by diagram. Express the result as a sum of similar diagrams, but with a new rule 1 that associates the expression ... with each dot, where λ = eβµ Now, with the awkward factors of N(N − 1) ∙ ∙ ∙ taken care of, you should find that the sum of all diagrams organizes itself into exponential form, resulting in the formula...Note that the exponent contains all connected diagrams, including those that can be disconnected by removal of a single line.(c) Using the properties of the grand partition function (see Problem), find diagrammatic expressions for the average number of particles and the pressure of this gas.(d) Keeping only the first diagram in each sum, express ... arid P(µ) in terms of an integral of the Mayer f-function. Eliminate µ to obtain the same result for the pressure (and the second virial coefficient) as derived in the text.(e) Repeat part (d) keeping the three-dot diagrams as well, to obtain an expression for the third virial coefficient in terms of an integral of f-functions. You should find that the Λ-shaped diagram cancels, leaving only the triangle diagram to contribute to C(T).Problem: In Section 6.5 I derived the useful relation F = − kT ln Z between the Helmholtz free energy and the ordinary partition function. Use an analogous argument to prove thatɸ = − κΤ ln Zwhere Z is the grand partition function and ɸ is the grand free energy introduced in Problem 1.Problem 1:By subtracting μN from U, H, F, or G, one can obtain four new thermodynamic potentials. Of the four, the most useful is the grand free energy (or grand potential),...(a) Derive the thermodynamic identity for Φ, and the related formulas for the partial derivatives of Φ with respect to T, V, and μ.(b) Prove that, for a system in thermal and diffusive equilibrium (with a reservoir that can supply both energy and particles), Φ tends to decrease.(c) Prove that Φ = – PV.(d) As a simple application, let the system he a single proton, which can be “occupied” either by a single electron (making a hydrogen atom, with energy –13.6 eV) of by none (with energy zero). Neglect the excited states of the atom and the two spin states of the electron, so that both the occupied and unoccupied states of the proton have zero entropy. Suppose that this proton is in the atmosphere of the sun, a reservoir with a temperature of 5800 K and an election concentration of about 2 × 1019 per cubic meter. Calculate Φ for both the occupied and unoccupied states, to determine which is more stable under these conditions. To compute the chemical potential of the electrons, treat them as an ideal gas. At about what temperature would the occupied and unoccupied states be equally stable, for this value of the electron concentration? (As in below Problem 2, the prediction for such a small system is only a probabilistic one.)Problem 2:The first excited energy level of a hydrogen atom has an energy of 10.2 eV, if wc take the ground-state energy to be zero. However, the first excited level is really four independent states, all with the same energy. We can therefore assign it an entropy of S = κ ln 4, since for this given value of the energy, the multiplicity is 4. Question: For what temperatures is the Helmholtz free energy of a hydrogen atom in the first excited level positive, and for what temperatures is it negative? (Comment: When F for the level is negative, the atom will spontaneously go from the ground state into that level, since F = 0 for the ground state and F always tends to decrease. However, for a system this small, the conclusion is only a probabilistic statement; random fluctuations will be very significant.) Get solution

15. For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four “neighbors” —above, below, left, and right. (Diagonal neighbors are normally not included.) What is the total energy (in terms of ϵ) for the particular state of the 4 × 4 square lattice shown in Figure?Figure:. One particular state of an Ising model on a 4 × 4 square lattice.... Get solution

16. Consider an Ising model of 100 elementary dipoles. Suppose you wish to calculate the partition function for this system, using a computer that can compute one billion terms of the partition function per second. How long must you wait for the answer? Get solution

17. Consider an Ising model of just two elementary dipoles, whose mutual interaction energy is ±ε. Enumerate the states of this system and write down their Boltzmann factors. Calculate the partition function. Find the probabilities of finding the dipoles parallel and antiparallel, and plot these probabilities as a function of κT/ε. Also calculate and plot the average energy of the system. At what temperatures are you more likely to find both dipoles pointing up than to find one up and one down? Get solution

18. Starting from the partition function, calculate the average energy of the one-dimensional Ising model, to verify equation. Sketch the average energy as a function of temperature.... Get solution

19. The critical temperature of iron is 1043 K. Use this value to make a rough estimate of the dipole-dipole interaction energy ϵ, in electron-volts. Get solution

20. Use a computer to plot ... as a function of κT/ϵ, as predicted by mean field theory, for a two-dimensional Ising model (with a square lattice). Get solution

21. At T = 0, equation says that ... Work out the first temperature-dependent correction to this value, in the limit βεn ≫ 1. Compare to the low-temperature behavior of a real ferromagnet, treated in Problem.Equation:...Problem:A ferromagnet is a material (like iron) that magnetizes spontaneously, even in the absence of an externally applied magnetic field. This happens because each elementary dipole has a strong tendency to align parallel to its neighbors. At T = 0 the magnetization of a ferromagnet has the maximum possible value, with all dipoles perfectly lined up; if there are N atoms, the total magnetization is typically ~2µBN, where µB is the Bohr magneton. At somewhat higher temperatures, the excitations take the form of spin waves, which can be visualized classically as shown in Figure. Like sound waves, spin waves are quantized: Each wave mode can have only integer multiples of a basic energy unit. In analogy with phonons, we think of the energy units as particles, called magnons. Each magnon reduces the total spin of the system by one unit of h/2π, and therefore reduces the magnetization by ~2µB. However, whereas the frequency of a sound wave is inversely proportional to its wavelength, the frequency of a spin wave is proportional to the square of 1/λ (in the limit of long wavelengths). Therefore, since ϵ = hf and p = h/λ for any “particle,” the energy of a magnon is proportional to the square of its momentum. In analogy with the energy-momentum relation for an ordinary nonrelativistic particle, we can write ϵ = p2/2m*, where m* is a constant related to the spin-spin interaction energy and the atomic spacing. For iron, m* turns out to equal 1.24 × 10−29 kg, about 14 times the mass of an electron. Another difference between magnons and phonons is that each magnon (or spin wave mode) has only one possible polarization.(a) Show that at low temperatures, the number of magnons per unit in a three-dimensional ferromagnet is given by...Evaluate the integral numerically.(b) Use the result of part (a) to find an expression for the fractional reduction in magnetization, (M(0) −M(T))/M(0). Write your answer in the form (T/T0)3/2, and estimate the constant T0 for iron.(c) Calculate the heat capacity due to magnetic excitations in a ferromagnet at low temperature. You should find Cy/Nk = (T/Ti)3/2, where TL differs from To only by a numerical constant. Estimate T1 for iron, and compare the magnon and phonon contributions to the heat capacity. (The Debye temperature of iron is 470 K.)(d) Consider a two-dimensional array of magnetic dipoles at low temperature. Assume that each elementary dipole can still point in any (three- dimensional) direction, so spin waves are still passible. Show that, the integral for the total number of magnons diverges in this case. (This result is an indication that there can be no spontaneous magnetization in such a two-dimensional system. However, in Section 8.2 we will consider a different two-dimensional model in which magnetization docs occur.)Figure: In the ground state of a ferromagnet, all the elementary dipoles point in the same direction. The lowest-energy excitations above the ground state are spin waves, in which the dipoles precess in a conical motion. A long-wavelength spin wave carries very little energy, because the difference in direction between neighboring dipoles is very small.... Get solution

22. Consider an Ising model in the presence of an external magnetic field B, which gives each dipole an additional energy of −µBB if it points up and +µBB if it points down (where fm is the dipole’s magnetic moment). Analyze this system using the mean field approximation to find the analogue of equation. Study the solutions of the equation graphically, and discuss the magnetization of this system as a function of both the external field strength and the temperature. Sketch the region in the T-B plane for which the equation has three solutions. Equation:... Get solution

23. The Ising model can be used to simulate other systems besides ferromagnets; examples include antiferromagnets, binary alloys, and even fluids. The Ising model of a fluid is called a lattice gas. We imagine that space is divided into a lattice of sites, each of which can be either occupied by a gas molecule or unoccupied. The system has no kinetic energy, and the only potential energy comes from interactions of molecules on adjacent sites. Specifically, there is a contribution of −u0 to the energy for each pair of neighboring sites that arc both occupied. (a) Write down a formula for the grand partition function for this system, as a function of u 0, T, and µ. (b) Rearrange your formula to show that it is identical, up to a multiplicative factor that does not depend on the state of the system, to the ordinary partition function for an Ising ferromagnet in the presence of an external magnetic field B, provided that you make the replacements u 0 → 4ϵ and µ → 2µBB −8ϵ. (Note that µ is the chemical potential of the gas while µB is the magnetic moment of a dipole in the magnet.)(c) Discuss the implications. Which states of the magnet correspond to low-density states of the lattice gas? Which states of the magnet correspond to high-density states in which the gas has condensed into a liquid? What shape does this model predict for the liquid-gas phase boundary in the P-T plane? Get solution

24. In this problem you will use the mean field approximation to analyze the behavior of the Ising model near the critical point.(a) Prove that, when x ≪ 1, ...(b) Use the result of part (a) to find an expression for the magnetization of the Ising model, in the mean field approximation, when T is very close to the critical temperature. You should find M α (Tc − T)β, where β (not to be confused with 1/κT) is a critical exponent, analogous to the β defined for a fluid in Problem. Onsager’s exact solution shows that β = 1/8 in two dimensions, while experiments and more sophisticated approximations show that β ≈ 1/3 in three dimensions. The mean field approximation, however, predicts a larger value.(c) The magnetic susceptibility x is defined as x = (∂M/∂B)T. The behavior of this quantity near the critical point is conventionally written as χ α (T − Tc)-γ, where γ is another critical exponent. Find the value of γ in the mean field approximation, and show that it does not depend on whether T is slightly above or slightly below Tc. (The exact value of γ in two dimensions turns out to be 7/4, while in three dimensions γ ≈ 1.24.)Probem:In this problem you will investigate the behavior of a van der Waals fluid near the critical point. It is easiest to work in terms of reduced variables throughout.(a) Expand the van der Waals equation in a Taylor series in (V − Vc) keeping terms through order (V − Vc)3. Argue that, for T sufficiently close to Tc, the term quadratic in (V − Vc) becomes negligible compared to the others and may be dropped.(b) The resulting expression for P(V) is antisymmetric about the point V = Vc Use this fact to find an approximate formula for the vapor pressure as a function of temperature. (You may find it helpful to plot the isotherm.) Evaluate the slope of the phase boundary, dP/dT, at the critical point.(c) Still working in the same limit, find an expression for the difference in volume between the gas and liquid phases at the vapor pressure. You should find (Vg −Vl) α (Tc −T)β, where β is known as a critical exponent. Experiments show that β has a universal value of about 1/3, but the van der Waals model predicts a larger value.(d) Use the previous result to calculate the predicted latent heat of the transformation as a function of temperature, and sketch this function.(e) The shape of the T − Tc isotherm defines another critical exponent, called δ: (P −Pc) α (V − Vc)δ. Calculate δ in the van der Waals model. (Experimental values of δ are typically around 4 or 5.)(f) A third critical exponent describes the temperature dependence of the isothermal compressibility,...This quantity diverges at the critical point, in proportion to a power of (T−Tc) that in principle could differ depending on Whether one approaches the critical point from above or below. Therefore the clinical exponents γ and γ’ are defined by the relations....Calculate k on both sides of the critical point in the van der Waals model, and show that γ = γ’ in this model. Get solution

25. In Problem you manually computed the energy of a particular state of a 4 × 4 square lattice. Repeat that computation, but this time apply periodic boundary conditions.Problem: For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four “neighbors” —above, below, left, and right. (Diagonal neighbors are normally not included.) What is the total energy (in terms of ϵ) for the particular state of the 4 × 4 square lattice shown in Figure?Figure:. One particular state of an Ising model on a 4 × 4 square lattice.... Get solution

26. Implement the ising program on your favorite computer, using your favorite programming language. Run it for various lattice sizes and temperatures and observe the results. In particular:(a) Run the program with a 20 × 20 lattice at T = 10, 5, 4, 3, and 2.5, for at least 100 iterations per dipole per run. At each temperature make a rough estimate of the size of the largest clusters.(b) Repeat part (a) for a 40 × 40 lattice. Are the cluster sizes any different? Explain.(c) Run the program with a 20 × 20 lattice at T = 2, 1.5, and 1. Estimate the average magnetization (as a percentage of total saturation) at each of these temperatures. Disregard runs in which the system gets stuck in a metastable state with two domains.(d) Run the program with a 10 × 10 lattice at T = 2.5. Watch it run for 100,000 iterations or so. Describe and explain the behavior.(e) Use successively larger lattices to estimate the typical cluster size at temperatures from 2.5 down to 2.27 (the critical temperature). The closer you arc to the critical temperature, the larger a lattice you’ll need and the longer the program will have to run. Quit when you realize that there are better ways to spend your time. Is it plausible that the cluster size goes to infinity as the temperature approaches the critical temperature? Get solution

27. Modify the ising program to compute the average energy of the system over all iterations. To do this, first add code to the initialize subroutine to compute the initial energy of the lattice; then, whenever a dipole is flipped, change the energy variable by the appropriate amount. When computing the average energy, be sure to average over all iterations, not just those iterations in which a dipole is actually flipped (why?). Run the program for a 5 × 5 lattice for T values from 4 down to 1 in reasonably small intervals, then plot the average energy as a function of T. Also plot the heat capacity. Use at least 1000 iterations per dipole for each run, preferably more. If your computer is fast enough, repeat for a 10 × 10 lattice and for a 20 × 20 lattice. Discuss the results. (Hint: Rather than starting over at each temperature with a random initial state, you can save time by starting with the final state generated at the previous, nearby temperature. For the larger lattices you may wish to save time by considering only a smaller temperature interval, perhaps from 3 down to 1.5.) Get solution

28. Modify the ising program to compute the total magnetization (that is, the sum of all the s values) for each iteration, and to tally how often each possible magnetization value occurs during a run, plotting the results as a histogram. Run the program for a 5 × 5 lattice at a variety of temperatures, and discuss the results. Sketch a graph of the most likely magnetization value as a function of temperature. If your computer is fast enough, repeat for a 10 × 10 lattice. Get solution

29. To quantify the clustering of alignments within an Ising magnet, we define a quantity called the correlation function, c(r). Take any two dipoles i and j, separated by a distance r, and compute the product of their states: sisj. This product is 1 if the dipoles are parallel and −1 if the dipoles are antiparallel. Now average this quantity over all pairs that are separated by a fixed distance r, to obtain a measure of the tendency of dipoles to be “correlated” over this distance. Finally, to remove the effect of any over all magnetization of the system, subtract off the square of the average s. Written as an equation, then, the correlation function is ...where it is understood that the first term averages over all pairs at the fixed distance r. Technically, the averages should also be taken over all possible states of the system, but don’t do this yet.(a) Add a routine to the ising program to compute the correlation function for the current state of the lattice, averaging over all pairs separated either vertically or horizontally (but not diagonally) by r units of distance, where r varies from 1 to half the lattice size. Have the program execute this routine periodically and plot the results as a bar graph.(b) Run this program at a variety of temperatures, above, below, and near the critical point. Use a lattice size of at least 20, preferably larger (especially near the critical point). Describe the behavior of the correlation function at each temperature.(c) Now add code to compute the average correlation function over the duration of a run. (However, it’s best to let the system “equilibrate” to a typical state before you begin accumulating averages.) The correlation length is defined as the distance over which the correlation function decreases by a factor of e. Estimate the correlation length at each temperature, and plot a graph of the correlation length vs. T. Get solution

30. Modifiy the ising program to simulate a one-dimensional Ising model.(a) For a lattice size of 100, observe the sequence of states generated at various temperatures and discuss the results. According to the exact solution (for an infinite lattice), we expect this system to magnetize only as the temperature goes to zero; is the behavior of your program consistent with this prediction ? Row does the typical cluster size depend on temperature?(b) Modify your program to compute the average energy as in Problem 1. Plot the energy and heat capacity vs. temperature and compare to the exact result for an infinite lattice.(c) Modify your program to compute the magnetization as in Problem 2. Determine the most likely magnetization for various temperatures, and discuss your results.Problem 1:Modify the ising program to compute the average energy of the system over all iterations. To do this, first add code to the initialize subroutine to compute the initial energy of the lattice; then, whenever a dipole is flipped, change the energy variable by the appropriate amount. When computing the average energy, be sure to average over all iterations, not just those iterations in which a dipole is actually flipped (why?). Run the program for a 5 × 5 lattice for T values from 4 down to 1 in reasonably small intervals, then plot the average energy as a function of T. Also plot the heat capacity. Use at least 1000 iterations per dipole for each run, preferably more. If your computer is fast enough, repeat for a 10 × 10 lattice and for a 20 × 20 lattice. Discuss the results. (Hint: Rather than starting over at each temperature with a random initial state, you can save time by starting with the final state generated at the previous, nearby temperature. For the larger lattices you may wish to save time by considering only a smaller temperature interval, perhaps from 3 down to 1.5.)Problem 2:Modify the ising program to compute the total magnetization (that is, the sum of all the s values) for each iteration, and to tally how often each possible magnetization value occurs during a run, plotting the results as a histogram. Run the program for a 5 × 5 lattice at a variety of temperatures, and discuss the results. Sketch a graph of the most likely magnetization value as a function of temperature. If your computer is fast enough, repeat for a 10 × 10 lattice. Get solution

31. Modify the ising program to simulate a three-dimensional Ising model with a simple cubic lattice. In whatever way you can, try to show that this system has a critical point at around T = 4.5. Get solution

32. Imagine taking a two-dimensional Ising lattice and dividing the sites into 3 × 3 “blocks”, as shown in Figure 8.11. In a block spin transformation, we replace the nine dipoles in each block with a single dipole, whose state is determined by “majority rule”: If more than half of the original dipoles point up, then the new dipole points up, while if more than half of the original dipoles point down, then the new dipole points down. By applying this transformation to the entire lattice, we reduce it to a new lattice whose width is 1/3 the original width. This transformation is one version of a renormalization group transformation, a powerful technique for studying the behavior of systems near their critical points. Get solution

2. Draw all the diagrams, connected or disconnected, representing terms in the configuration integral with four factors of fij. You should find 11 diagrams in total, of which five are connected. Get solution

3. Keeping only the first two diagrams in equation , and approximating N ≈ N − l ≈ N − 2 ≈ ∙ ∙ ∙, expand the exponential in a power series through the third power. Multiply each term out, and show that all the numerical coefficients give precisely the correct symmetry factors for the disconnected diagrams.Equation:... Get solution

4. Draw all the connected diagrams containing four dots. There are six diagrams in total; be careful to avoid drawing two diagrams that look superficially different but are actually the same. Which of the diagrams would remain connected if any single dot were removed? Get solution

5. By changing variables as in the text, express the diagram in equation 8.18 in terms of the same integral as in equation 8.31. Do the same for the last two diagrams in the first line of equation. Which diagrams cannot be written in terms of this basic integral?Equation:... Get solution

6. You can estimate the size of any diagram by realizing that f(r) is of order 1 out to a distance of about the diameter of a molecule, and f ≈ 0 beyond that. Hence, a three-dimensional integral of a product of f’s will generally give a result that is of the order of the volume of a molecule. Estimate the sizes of all the diagrams shown explicitly in equation, and explain why it was necessary to rewrite the series in exponential form. Equation:... Get solution

7. Show that, if you don’t make too many approximations, the exponential series in equation includes the three-dot diagram in equation 8.18. There will be some leftover terms; show that these vanish in the thermodynamic limit.Equation:... Get solution

8. Show that the nth virial coefficient depends on the diagrams in equation that have n dots. Write the third virial coefficient, C(T), in terms of an integral of f-functions. Why it would be difficult to carry out this integral?Equation:... Get solution

9. Show that the Lennard-Jones potential reaches its minimum value at r = r0, and that its value at this minimum is –u0 At what value of r does the potential equal zero? Get solution

10. Use a computer to calculate and plot the second virial coefficient for a gas of molecules interacting via the Lennard-Jones potential, for values of κT/u0 ranging from 1 to 7. On the same graph, plot the data for nitrogen given in Problem, choosing the parameters r0 and u0 so as to obtain a good fit.Problem:Even at low density, real gases don’t quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial expansion,...where the functions B(T), C(T), and so on are called the virial coefficients. When the density of the gas is fairly low, so that the volume per mole is large, each term in the series is much smaller than the one before. In many situations it’s sufficient to omit the third term and concentrate on the second, whose coefficient B(T) is called the second virial coefficient (the first coefficient being 1). Here are some measured values of the second virial coefficient for nitrogen (N2): T (K)B (cm3/mol)100−160200− 35300− 4.24009.050016.960021.3(a) For each temperature in the table, compute the second term in the virial equation, B(T)/(V/n), for nitrogen at atmospheric pressure. Discuss the validity of the ideal gas law under these conditions.(b) Think about the forces between molecules, and explain why we might expect B(T) to be negative at low temperatures but positive at high temperatures.(c) Any proposed relation between P, V, and T , like the ideal gas law or the virial equation, is called an equation of state. Another famous equation of state, which is qualitatively accurate even for dense fluids, is the van der Waals equation,...where a and b are constants that depend on the type of gas. Calculate the second and third virial coefficients (B and C) for a gas obeying the van der Waals equation, in terms of a and b. (Hint: The binomial expansion says that ... provided that |px| ≪ 1. Apply this approximation to the quantity [1 − (nb/V)]−1.)(d) Plot a graph of the van der Waals prediction for B(T), choosing a and b so as to approximately match the data given above for nitrogen. Discuss the accuracy of the van der Waals equation over this range of conditions. (The van der Waals equation is discussed much further in Section 5.3.) Get solution

11. Consider a gas of “hard spheres,” which do not interact at all unless their separation distance is less than r0, in which case then interaction energy is infinite. Sketch the Mayer f-function for this gas, and compute the second virial coefficient. Discuss the result briefly. Get solution

12. Consider a gas of molecules whose interaction energy u(r) is infinite for r r0 and negative for r > r0, with a minimum value of –u0. Suppose further that κT ≫ u0, so you can approximate the Boltzmann factor for r > r0 using ex ≈ 1 + x. Show that under these conditions the second virial coefficient has the form B(T) = b − (a/κT), the same as what you found for a van der Waals gas in Problem. Write the van der Waals constants a and b in terms of r0 and u(r), and discuss the results briefly. Problem:Even at low density, real gases don’t quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial expansion,...where the functions B(T), C(T), and so on are called the virial coefficients. When the density of the gas is fairly low, so that the volume per mole is large, each term in the series is much smaller than the one before. In many situations it’s sufficient to omit the third term and concentrate on the second, whose coefficient B(T) is called the second virial coefficient (the first coefficient being 1). Here are some measured values of the second virial coefficient for nitrogen (N2): T (K)B (cm3/mol)100−160200− 35300− 4.24009.050016.960021.3(a) For each temperature in the table, compute the second term in the virial equation, B(T)/(V/n), for nitrogen at atmospheric pressure. Discuss the validity of the ideal gas law under these conditions.(b) Think about the forces between molecules, and explain why we might expect B(T) to be negative at low temperatures but positive at high temperatures.(c) Any proposed relation between P, V, and T , like the ideal gas law or the virial equation, is called an equation of state. Another famous equation of state, which is qualitatively accurate even for dense fluids, is the van der Waals equation,...where a and b are constants that depend on the type of gas. Calculate the second and third virial coefficients (B and C) for a gas obeying the van der Waals equation, in terms of a and b. (Hint: The binomial expansion says that ... provided that |px| ≪ 1. Apply this approximation to the quantity [1 − (nb/V)]−1.)(d) Plot a graph of the van der Waals prediction for B(T), choosing a and b so as to approximately match the data given above for nitrogen. Discuss the accuracy of the van der Waals equation over this range of conditions. (The van der Waals equation is discussed much further in Section 5.3.) Get solution

13. Use the cluster expansion to write the total energy of a monatomic nonideal gas in terms of a sum of diagrams. Keeping only the first diagram, show that the energy is approximately...Use a computer to evaluate this integral numerically, as a function of T, for the Lennard-Jones potential. Plot the temperature-dependent part of the correction term, and explain the shape of the graph physically. Discuss the correction to the heat capacity at constant volume, and compute this correction numerically for argon at room temperature and atmospheric pressure. Get solution

14. In this section I’ve formulated the cluster expansion for a gas with a fixed number of particles, using the “canonical” formalism of Chapter 6. A somewhat cleaner approach, however, is to use the “grand canonical” formalism introduced in Section 7.1, in which we allow the system to exchange particles with a much larger reservoir.(a) Write down a formula for the grand partition function (Z) of a weakly interacting gas in thermal and diffusive equilibrium with a reservoir at fixed T and µ. Express Z as a sum over all possible particle numbers N, with each term involving the ordinary partition function Z(N).(b) Use equations to express Z(N) as a sum of diagrams, then carry out the sum over N, diagram by diagram. Express the result as a sum of similar diagrams, but with a new rule 1 that associates the expression ... with each dot, where λ = eβµ Now, with the awkward factors of N(N − 1) ∙ ∙ ∙ taken care of, you should find that the sum of all diagrams organizes itself into exponential form, resulting in the formula...Note that the exponent contains all connected diagrams, including those that can be disconnected by removal of a single line.(c) Using the properties of the grand partition function (see Problem), find diagrammatic expressions for the average number of particles and the pressure of this gas.(d) Keeping only the first diagram in each sum, express ... arid P(µ) in terms of an integral of the Mayer f-function. Eliminate µ to obtain the same result for the pressure (and the second virial coefficient) as derived in the text.(e) Repeat part (d) keeping the three-dot diagrams as well, to obtain an expression for the third virial coefficient in terms of an integral of f-functions. You should find that the Λ-shaped diagram cancels, leaving only the triangle diagram to contribute to C(T).Problem: In Section 6.5 I derived the useful relation F = − kT ln Z between the Helmholtz free energy and the ordinary partition function. Use an analogous argument to prove thatɸ = − κΤ ln Zwhere Z is the grand partition function and ɸ is the grand free energy introduced in Problem 1.Problem 1:By subtracting μN from U, H, F, or G, one can obtain four new thermodynamic potentials. Of the four, the most useful is the grand free energy (or grand potential),...(a) Derive the thermodynamic identity for Φ, and the related formulas for the partial derivatives of Φ with respect to T, V, and μ.(b) Prove that, for a system in thermal and diffusive equilibrium (with a reservoir that can supply both energy and particles), Φ tends to decrease.(c) Prove that Φ = – PV.(d) As a simple application, let the system he a single proton, which can be “occupied” either by a single electron (making a hydrogen atom, with energy –13.6 eV) of by none (with energy zero). Neglect the excited states of the atom and the two spin states of the electron, so that both the occupied and unoccupied states of the proton have zero entropy. Suppose that this proton is in the atmosphere of the sun, a reservoir with a temperature of 5800 K and an election concentration of about 2 × 1019 per cubic meter. Calculate Φ for both the occupied and unoccupied states, to determine which is more stable under these conditions. To compute the chemical potential of the electrons, treat them as an ideal gas. At about what temperature would the occupied and unoccupied states be equally stable, for this value of the electron concentration? (As in below Problem 2, the prediction for such a small system is only a probabilistic one.)Problem 2:The first excited energy level of a hydrogen atom has an energy of 10.2 eV, if wc take the ground-state energy to be zero. However, the first excited level is really four independent states, all with the same energy. We can therefore assign it an entropy of S = κ ln 4, since for this given value of the energy, the multiplicity is 4. Question: For what temperatures is the Helmholtz free energy of a hydrogen atom in the first excited level positive, and for what temperatures is it negative? (Comment: When F for the level is negative, the atom will spontaneously go from the ground state into that level, since F = 0 for the ground state and F always tends to decrease. However, for a system this small, the conclusion is only a probabilistic statement; random fluctuations will be very significant.) Get solution

15. For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four “neighbors” —above, below, left, and right. (Diagonal neighbors are normally not included.) What is the total energy (in terms of ϵ) for the particular state of the 4 × 4 square lattice shown in Figure?Figure:. One particular state of an Ising model on a 4 × 4 square lattice.... Get solution

16. Consider an Ising model of 100 elementary dipoles. Suppose you wish to calculate the partition function for this system, using a computer that can compute one billion terms of the partition function per second. How long must you wait for the answer? Get solution

17. Consider an Ising model of just two elementary dipoles, whose mutual interaction energy is ±ε. Enumerate the states of this system and write down their Boltzmann factors. Calculate the partition function. Find the probabilities of finding the dipoles parallel and antiparallel, and plot these probabilities as a function of κT/ε. Also calculate and plot the average energy of the system. At what temperatures are you more likely to find both dipoles pointing up than to find one up and one down? Get solution

18. Starting from the partition function, calculate the average energy of the one-dimensional Ising model, to verify equation. Sketch the average energy as a function of temperature.... Get solution

19. The critical temperature of iron is 1043 K. Use this value to make a rough estimate of the dipole-dipole interaction energy ϵ, in electron-volts. Get solution

20. Use a computer to plot ... as a function of κT/ϵ, as predicted by mean field theory, for a two-dimensional Ising model (with a square lattice). Get solution

21. At T = 0, equation says that ... Work out the first temperature-dependent correction to this value, in the limit βεn ≫ 1. Compare to the low-temperature behavior of a real ferromagnet, treated in Problem.Equation:...Problem:A ferromagnet is a material (like iron) that magnetizes spontaneously, even in the absence of an externally applied magnetic field. This happens because each elementary dipole has a strong tendency to align parallel to its neighbors. At T = 0 the magnetization of a ferromagnet has the maximum possible value, with all dipoles perfectly lined up; if there are N atoms, the total magnetization is typically ~2µBN, where µB is the Bohr magneton. At somewhat higher temperatures, the excitations take the form of spin waves, which can be visualized classically as shown in Figure. Like sound waves, spin waves are quantized: Each wave mode can have only integer multiples of a basic energy unit. In analogy with phonons, we think of the energy units as particles, called magnons. Each magnon reduces the total spin of the system by one unit of h/2π, and therefore reduces the magnetization by ~2µB. However, whereas the frequency of a sound wave is inversely proportional to its wavelength, the frequency of a spin wave is proportional to the square of 1/λ (in the limit of long wavelengths). Therefore, since ϵ = hf and p = h/λ for any “particle,” the energy of a magnon is proportional to the square of its momentum. In analogy with the energy-momentum relation for an ordinary nonrelativistic particle, we can write ϵ = p2/2m*, where m* is a constant related to the spin-spin interaction energy and the atomic spacing. For iron, m* turns out to equal 1.24 × 10−29 kg, about 14 times the mass of an electron. Another difference between magnons and phonons is that each magnon (or spin wave mode) has only one possible polarization.(a) Show that at low temperatures, the number of magnons per unit in a three-dimensional ferromagnet is given by...Evaluate the integral numerically.(b) Use the result of part (a) to find an expression for the fractional reduction in magnetization, (M(0) −M(T))/M(0). Write your answer in the form (T/T0)3/2, and estimate the constant T0 for iron.(c) Calculate the heat capacity due to magnetic excitations in a ferromagnet at low temperature. You should find Cy/Nk = (T/Ti)3/2, where TL differs from To only by a numerical constant. Estimate T1 for iron, and compare the magnon and phonon contributions to the heat capacity. (The Debye temperature of iron is 470 K.)(d) Consider a two-dimensional array of magnetic dipoles at low temperature. Assume that each elementary dipole can still point in any (three- dimensional) direction, so spin waves are still passible. Show that, the integral for the total number of magnons diverges in this case. (This result is an indication that there can be no spontaneous magnetization in such a two-dimensional system. However, in Section 8.2 we will consider a different two-dimensional model in which magnetization docs occur.)Figure: In the ground state of a ferromagnet, all the elementary dipoles point in the same direction. The lowest-energy excitations above the ground state are spin waves, in which the dipoles precess in a conical motion. A long-wavelength spin wave carries very little energy, because the difference in direction between neighboring dipoles is very small.... Get solution

22. Consider an Ising model in the presence of an external magnetic field B, which gives each dipole an additional energy of −µBB if it points up and +µBB if it points down (where fm is the dipole’s magnetic moment). Analyze this system using the mean field approximation to find the analogue of equation. Study the solutions of the equation graphically, and discuss the magnetization of this system as a function of both the external field strength and the temperature. Sketch the region in the T-B plane for which the equation has three solutions. Equation:... Get solution

23. The Ising model can be used to simulate other systems besides ferromagnets; examples include antiferromagnets, binary alloys, and even fluids. The Ising model of a fluid is called a lattice gas. We imagine that space is divided into a lattice of sites, each of which can be either occupied by a gas molecule or unoccupied. The system has no kinetic energy, and the only potential energy comes from interactions of molecules on adjacent sites. Specifically, there is a contribution of −u0 to the energy for each pair of neighboring sites that arc both occupied. (a) Write down a formula for the grand partition function for this system, as a function of u 0, T, and µ. (b) Rearrange your formula to show that it is identical, up to a multiplicative factor that does not depend on the state of the system, to the ordinary partition function for an Ising ferromagnet in the presence of an external magnetic field B, provided that you make the replacements u 0 → 4ϵ and µ → 2µBB −8ϵ. (Note that µ is the chemical potential of the gas while µB is the magnetic moment of a dipole in the magnet.)(c) Discuss the implications. Which states of the magnet correspond to low-density states of the lattice gas? Which states of the magnet correspond to high-density states in which the gas has condensed into a liquid? What shape does this model predict for the liquid-gas phase boundary in the P-T plane? Get solution

24. In this problem you will use the mean field approximation to analyze the behavior of the Ising model near the critical point.(a) Prove that, when x ≪ 1, ...(b) Use the result of part (a) to find an expression for the magnetization of the Ising model, in the mean field approximation, when T is very close to the critical temperature. You should find M α (Tc − T)β, where β (not to be confused with 1/κT) is a critical exponent, analogous to the β defined for a fluid in Problem. Onsager’s exact solution shows that β = 1/8 in two dimensions, while experiments and more sophisticated approximations show that β ≈ 1/3 in three dimensions. The mean field approximation, however, predicts a larger value.(c) The magnetic susceptibility x is defined as x = (∂M/∂B)T. The behavior of this quantity near the critical point is conventionally written as χ α (T − Tc)-γ, where γ is another critical exponent. Find the value of γ in the mean field approximation, and show that it does not depend on whether T is slightly above or slightly below Tc. (The exact value of γ in two dimensions turns out to be 7/4, while in three dimensions γ ≈ 1.24.)Probem:In this problem you will investigate the behavior of a van der Waals fluid near the critical point. It is easiest to work in terms of reduced variables throughout.(a) Expand the van der Waals equation in a Taylor series in (V − Vc) keeping terms through order (V − Vc)3. Argue that, for T sufficiently close to Tc, the term quadratic in (V − Vc) becomes negligible compared to the others and may be dropped.(b) The resulting expression for P(V) is antisymmetric about the point V = Vc Use this fact to find an approximate formula for the vapor pressure as a function of temperature. (You may find it helpful to plot the isotherm.) Evaluate the slope of the phase boundary, dP/dT, at the critical point.(c) Still working in the same limit, find an expression for the difference in volume between the gas and liquid phases at the vapor pressure. You should find (Vg −Vl) α (Tc −T)β, where β is known as a critical exponent. Experiments show that β has a universal value of about 1/3, but the van der Waals model predicts a larger value.(d) Use the previous result to calculate the predicted latent heat of the transformation as a function of temperature, and sketch this function.(e) The shape of the T − Tc isotherm defines another critical exponent, called δ: (P −Pc) α (V − Vc)δ. Calculate δ in the van der Waals model. (Experimental values of δ are typically around 4 or 5.)(f) A third critical exponent describes the temperature dependence of the isothermal compressibility,...This quantity diverges at the critical point, in proportion to a power of (T−Tc) that in principle could differ depending on Whether one approaches the critical point from above or below. Therefore the clinical exponents γ and γ’ are defined by the relations....Calculate k on both sides of the critical point in the van der Waals model, and show that γ = γ’ in this model. Get solution

25. In Problem you manually computed the energy of a particular state of a 4 × 4 square lattice. Repeat that computation, but this time apply periodic boundary conditions.Problem: For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four “neighbors” —above, below, left, and right. (Diagonal neighbors are normally not included.) What is the total energy (in terms of ϵ) for the particular state of the 4 × 4 square lattice shown in Figure?Figure:. One particular state of an Ising model on a 4 × 4 square lattice.... Get solution

26. Implement the ising program on your favorite computer, using your favorite programming language. Run it for various lattice sizes and temperatures and observe the results. In particular:(a) Run the program with a 20 × 20 lattice at T = 10, 5, 4, 3, and 2.5, for at least 100 iterations per dipole per run. At each temperature make a rough estimate of the size of the largest clusters.(b) Repeat part (a) for a 40 × 40 lattice. Are the cluster sizes any different? Explain.(c) Run the program with a 20 × 20 lattice at T = 2, 1.5, and 1. Estimate the average magnetization (as a percentage of total saturation) at each of these temperatures. Disregard runs in which the system gets stuck in a metastable state with two domains.(d) Run the program with a 10 × 10 lattice at T = 2.5. Watch it run for 100,000 iterations or so. Describe and explain the behavior.(e) Use successively larger lattices to estimate the typical cluster size at temperatures from 2.5 down to 2.27 (the critical temperature). The closer you arc to the critical temperature, the larger a lattice you’ll need and the longer the program will have to run. Quit when you realize that there are better ways to spend your time. Is it plausible that the cluster size goes to infinity as the temperature approaches the critical temperature? Get solution

27. Modify the ising program to compute the average energy of the system over all iterations. To do this, first add code to the initialize subroutine to compute the initial energy of the lattice; then, whenever a dipole is flipped, change the energy variable by the appropriate amount. When computing the average energy, be sure to average over all iterations, not just those iterations in which a dipole is actually flipped (why?). Run the program for a 5 × 5 lattice for T values from 4 down to 1 in reasonably small intervals, then plot the average energy as a function of T. Also plot the heat capacity. Use at least 1000 iterations per dipole for each run, preferably more. If your computer is fast enough, repeat for a 10 × 10 lattice and for a 20 × 20 lattice. Discuss the results. (Hint: Rather than starting over at each temperature with a random initial state, you can save time by starting with the final state generated at the previous, nearby temperature. For the larger lattices you may wish to save time by considering only a smaller temperature interval, perhaps from 3 down to 1.5.) Get solution

28. Modify the ising program to compute the total magnetization (that is, the sum of all the s values) for each iteration, and to tally how often each possible magnetization value occurs during a run, plotting the results as a histogram. Run the program for a 5 × 5 lattice at a variety of temperatures, and discuss the results. Sketch a graph of the most likely magnetization value as a function of temperature. If your computer is fast enough, repeat for a 10 × 10 lattice. Get solution

29. To quantify the clustering of alignments within an Ising magnet, we define a quantity called the correlation function, c(r). Take any two dipoles i and j, separated by a distance r, and compute the product of their states: sisj. This product is 1 if the dipoles are parallel and −1 if the dipoles are antiparallel. Now average this quantity over all pairs that are separated by a fixed distance r, to obtain a measure of the tendency of dipoles to be “correlated” over this distance. Finally, to remove the effect of any over all magnetization of the system, subtract off the square of the average s. Written as an equation, then, the correlation function is ...where it is understood that the first term averages over all pairs at the fixed distance r. Technically, the averages should also be taken over all possible states of the system, but don’t do this yet.(a) Add a routine to the ising program to compute the correlation function for the current state of the lattice, averaging over all pairs separated either vertically or horizontally (but not diagonally) by r units of distance, where r varies from 1 to half the lattice size. Have the program execute this routine periodically and plot the results as a bar graph.(b) Run this program at a variety of temperatures, above, below, and near the critical point. Use a lattice size of at least 20, preferably larger (especially near the critical point). Describe the behavior of the correlation function at each temperature.(c) Now add code to compute the average correlation function over the duration of a run. (However, it’s best to let the system “equilibrate” to a typical state before you begin accumulating averages.) The correlation length is defined as the distance over which the correlation function decreases by a factor of e. Estimate the correlation length at each temperature, and plot a graph of the correlation length vs. T. Get solution

30. Modifiy the ising program to simulate a one-dimensional Ising model.(a) For a lattice size of 100, observe the sequence of states generated at various temperatures and discuss the results. According to the exact solution (for an infinite lattice), we expect this system to magnetize only as the temperature goes to zero; is the behavior of your program consistent with this prediction ? Row does the typical cluster size depend on temperature?(b) Modify your program to compute the average energy as in Problem 1. Plot the energy and heat capacity vs. temperature and compare to the exact result for an infinite lattice.(c) Modify your program to compute the magnetization as in Problem 2. Determine the most likely magnetization for various temperatures, and discuss your results.Problem 1:Modify the ising program to compute the average energy of the system over all iterations. To do this, first add code to the initialize subroutine to compute the initial energy of the lattice; then, whenever a dipole is flipped, change the energy variable by the appropriate amount. When computing the average energy, be sure to average over all iterations, not just those iterations in which a dipole is actually flipped (why?). Run the program for a 5 × 5 lattice for T values from 4 down to 1 in reasonably small intervals, then plot the average energy as a function of T. Also plot the heat capacity. Use at least 1000 iterations per dipole for each run, preferably more. If your computer is fast enough, repeat for a 10 × 10 lattice and for a 20 × 20 lattice. Discuss the results. (Hint: Rather than starting over at each temperature with a random initial state, you can save time by starting with the final state generated at the previous, nearby temperature. For the larger lattices you may wish to save time by considering only a smaller temperature interval, perhaps from 3 down to 1.5.)Problem 2:Modify the ising program to compute the total magnetization (that is, the sum of all the s values) for each iteration, and to tally how often each possible magnetization value occurs during a run, plotting the results as a histogram. Run the program for a 5 × 5 lattice at a variety of temperatures, and discuss the results. Sketch a graph of the most likely magnetization value as a function of temperature. If your computer is fast enough, repeat for a 10 × 10 lattice. Get solution

31. Modify the ising program to simulate a three-dimensional Ising model with a simple cubic lattice. In whatever way you can, try to show that this system has a critical point at around T = 4.5. Get solution

32. Imagine taking a two-dimensional Ising lattice and dividing the sites into 3 × 3 “blocks”, as shown in Figure 8.11. In a block spin transformation, we replace the nine dipoles in each block with a single dipole, whose state is determined by “majority rule”: If more than half of the original dipoles point up, then the new dipole points up, while if more than half of the original dipoles point down, then the new dipole points down. By applying this transformation to the entire lattice, we reduce it to a new lattice whose width is 1/3 the original width. This transformation is one version of a renormalization group transformation, a powerful technique for studying the behavior of systems near their critical points. Get solution