/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 A magnet can be modeled as an en... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 7

A magnet can be modeled as an enormous collection of electronic spins. In the simplest model, known as the Ising model, the spins can point only up or down, and are assigned the values \(S_{i}=\pm 1\), for \(l=1, \ldots, N>>1 .\) For quantum mechanical reasons, the spins like to point in the same direction as their neighbors; on the other hand, the randomizing effects of temperature tend to disrupt any such alignment. An important macroscopic property of the magnet is its average spin or magnetization $$ m=\left|\frac{I}{N} \sum_{i=1}^{N} S_{i}\right| $$ At high temperature the spins point in random directions and so \(m \approx 0 ;\) the material is in the paramagnetic state. As the temperature is lowered, \(m\) remains near zero until a critical temperature \(T_{e}\) is reached. Then a phase transition occurs and the material spontaneously magnetizes. Now \(m>0 ;\) we have a ferromagnet. But the symmetry between up and down spins means that there are two possible ferromagnetic states. This symmetry can be broken by applying an external magnetic field \(h\), which favors either the up or down direction. Then, in an approximation called mean-field theory, the equation governing the equilibrium value of \(m\) is $$ h=T \tanh ^{-1} \mathrm{~m}-\mathrm{Jnm} $$ where \(J\) and \(n\) are constants; \(J>0\) is the ferromagnetic coupling strength and \(n\) is the number of neighbors of each spin (Ma 1985, p. 459 ). a) Analyze the solutions \(m^{*}\) of \(h=T \tanh ^{-1} m-I n m\), using a graphical approach. b) For the special case \(h=0\), find the critical temperature \(T_{c}\) at which a phase transition occurs.

Short Answer

Expert verified
In summary, to analyze the solutions of the equilibrium equation \(h=T \tanh^{-1}m - Jnm\) graphically, plot both sides of the equation against \(m\). Find the intersections points on the graph, which represent the equilibrium magnetizations \(m^*\). For the special case when \(h=0\), equate the derivatives of the two functions with respect to \(m\) and solve for the critical temperature \(T_c\), at which a phase transition occurs.

Step by step solution

01

Graphical representation of the equilibrium equation

To analyze the solutions of the equilibrium equation, we need to plot both sides of the equation against \(m\). On the y-axis, we will have \(h\) and on the x-axis, we will have \(m\). We know that the equilibrium equation is given by: \(h = T \tanh^{-1}m - Jnm\)
02

Find the intersection points on the graph

The equilibrium magnetization \(m^*\) corresponds to the points where the two curves intersect on the graph. To find these points, plot the function \(h = T \tanh^{-1}m\) and the function \(h = Jnm\) separately and then identify the intersection points. At these intersection points, the curves satisfy the equilibrium equation. The corresponding magnetization values at these points are the equilibrium magnetization values \(m^*\).
03

Analyze the case when h=0 to find the critical temperature

Now, we need to work with the special case when \(h=0\). In this case, the equilibrium equation becomes: \(0 = T \tanh^{-1}m - Jnm\) We need to find the critical temperature \(T_c\) at which a phase transition occurs. To do this, we can analyze the behavior of the equation for different values of \(T\). To find the critical temperature, we can set the derivative of the function \(T \tanh^{-1}m\) with respect to \(m\) equal to the derivative of the function \(Jnm\) with respect to \(m\). So, we will have: \(\frac{dT \tanh^{-1}m}{dm} = \frac{dJnm}{dm}\) Compute these derivatives and solve for \(T\). The value obtained will be the critical temperature at which a phase transition occurs, \(T_c\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Magnetization
Magnetization is a crucial concept in understanding the fundamental behavior of magnetic materials. It signifies the degree to which a material can be magnetized, which, in simple terms, relates to the net amount of magnetic moment per unit volume. In the context of the Ising model, magnetization is expressed as the absolute value of the average spin:

\[ m = \left|\frac{1}{N} \sum_{i=1}^{N} S_{i}\right| \]

Here, the spin states, denoted by \(S_i\), can either be up (+1) or down (-1). Magnetization changes with varying temperatures. At high temperatures, spins are random due to thermal agitation, resulting in low magnetization. As the temperature decreases, spins start aligning, increasing the magnetization until a phase transition point where a dramatic change can be observed.
Phase Transition
A phase transition is a transformation in the state of a system, generally implying a change in some structural attributes. Within magnetic systems, the phase transition from a paramagnetic to a ferromagnetic state upon cooling is of particular interest. Specifically, the Ising model predicts that at a high temperature (above the critical temperature \(T_c\)), a material will be in a disordered paramagnetic state with virtually no net magnetization. However, as the temperature falls below \(T_c\), a spontaneous phase transition occurs, and the material attains a substantial net magnetization, marking the onset of the ferromagnetic state. The intensity of this change is rooted in the interactions between neighboring spins and the prevailing temperature.
Mean-Field Theory
Mean-field theory is a mathematical approach used to approximate the behavior of complex systems by considering the effect of all other individual components as an average effect on any single component. Applying mean-field theory to the Ising model simplifies the complexity by assuming each spin interacts with an average field due to neighboring spins rather than each neighbor individually.

The mean-field approximation gives rise to an equilibrium equation, which looks like: \[ h=T \tanh ^{-1} m-Jnm \]

Where \(h\) represents an external magnetic field, \(T\) the temperature, \(m\) the magnetization, \(J\) the coupling constant indicative of interaction strength between spins, and \(n\) the number of neighboring spins. This equation helps us determine how magnetization behaves under different external conditions.
Critical Temperature
Critical temperature, or \(T_c\), is a pivotal concept in the study of phase transitions. It's the temperature at which a magnetic or thermal phase transition occurs; for magnetic systems, it's where the material changes from paramagnetic to ferromagnetic upon cooling. In the Ising model, the critical temperature marks the point at which spontaneous magnetization starts to appear due to the alignment of spins within the material. This characteristic temperature is unique to each material and is determined by the interplay between thermal energy and magnetic interactions (quantified by the coupling constant \(J\) and the number of interacting neighbors \(n\)). To find \(T_c\), we look for the temperature at which the function's derivative, representing our system's energy, equals that of its magnetization.
Ferromagnetic State
The ferromagnetic state is a magnetic phase wherein individual spins within a material are aligned, resulting in a strong net magnetization. This state is in stark contrast to the paramagnetic phase, where spins are disorganized and hence the net magnetization is negligible. In the ferromagnetic state, spins align either in the presence of an external magnetic field or spontaneously when the material is below its critical temperature \(T_c\), which is a hallmark feature of a ferromagnetic phase transition.

The analysis of the Ising model demonstrates how a zero external magnetic field \(h=0\) and temperature decrease lead to this substantial magnetic order. The model implies that in the ferromagnetic state, an initial small alignment due to an external field can grow to result in a significant magnetization of the entire material.

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Most popular questions from this chapter

In pioneering work in epidemiology, Kermack and McKendrick ( 1927 ) proposed the following simple model for the evolution of an epidemic. Suppose that the population can be divided into three classes: \(x(t)=\) number of healthy people; \(y(t)=\) number of sick people; \(z(t)=\) number of dead people. Assume that the total population remains constant in size, except for deaths due to the epidemic. (That is, the epidemic evolves so rapidly that we can ignore the slower changes in the populations due to births, emigration, or deaths by other causes.) Then the model is $$ \begin{aligned} &\dot{x}=-k x y \\ &\dot{y}=k x y-\ell y \\ &\dot{z}=\ell y \end{aligned} $$ where \(k\) and \(\ell\) are positive constants. The equations are based on two assumptions: (i) Healthy people get sick at a rate proportional to the product of \(x\) and \(y\). This would be true if healthy and sick people encounter each other at a rate proportional to their numbers, and if there were a constant probability that each such encounter would lead to transmission of the disease. (ii) Sick people die at a constant rate \(\ell\). The goal of this exercise is to reduce the model, which is a third-order system, to a first-order system that can analyzed by our methods. (In Chapter 6 we will see a simpler analysis.) a) Show that \(x+y+z=N\), where \(N\) is constant. b) Use the \(\dot{x}\) and \(\dot{z}\) equation to show that \(x(t)=x_{0} \exp (-k z(t) / \ell)\), where \(x_{0}=x(0)\) c) Show that \(z\) satisfies the first-order equation \(\dot{z}=\ell\left[N-z- x_{0} \exp (-k z / \ell)\right]\). d) Show that this equation can be nondimensionalized to $$ \frac{d u}{d \tau}=a-b u-e^{-4} $$ by an appropriate rescaling. e) Show that \(a \geq 1\) and \(b>0\). f) Determine the number of fixed points \(u^{*}\) and classify their stability. g) Show that the maximum of \(\dot{u}(t)\) occurs at the same time as the maximum of both \(\tilde{z}(t)\) and \(y(t)\), (This time is called the peak of the epidemic, denoted \(t_{\text {peak }}\). At this time, there are more sick people and a higher daily death rate than at any other time.) h) Show that if \(b<1\), then \(\dot{u}(t)\) is increasing at \(t=0\) and reaches its maximum at some time \(t_{\text {peak }}>0\). Thus things get worse before they get better. (The term epidemic is reserved for this case.) Show that \(\dot{u}(t)\) eventually decreases to 0 . i) On the other hand, show that \(t_{\text {peak }}=0\) if \(b>1\). (Hence no epidemic occurs if \(b>1 .\) j) The condition \(b=1\) is the threshold condition for an epidemic to occur. Can you give a biological interpretation of this condition? k) Kermack and McKendrick showed that their model gave a good fit to data from the Bombay plague of 1906 . How would you improve the model to make it more appropriate for AIDS? Which assumptions need revising? For an introduction to models of epidemics, see Murray (1989), Chapter 19, or Edelstein-Keshet (1988). Models of AIDS are discussed by Murray (1989) and May and Anderson (1987). An excellent review and commentary on the Kermack- McKendrick papers is given by Anderson (1991).

A refinement of the model in the last exercise is $$ \dot{N}=r N\left(1-\frac{N}{K}\right)-H \frac{N}{A+N} $$ where \(H>0\) and \(A>0\). This model is more realistic in two respects: it has a fixed point at \(N=0\) for all values of the parameters, and the rate at which fish are caught decreases with \(N\). This is plausible-when fewer fish are available, it is harder to find them and so the daily catch drops. a) Give a biological interpretation of the parameter \(A ;\) what does it measure? b) Show that the system can be rewritten in dimensionless form as $$ \frac{d x}{d \tau}=x(1-x)-h \frac{x}{a+x} $$ for suitably defined dimensionless quantities \(x, \tau, a\), and \(h\). c) Show that the system can have one, two, or three fixed points, depending on the values of \(a\) and \(h\), Classify the stability of the fixed points in each case. d) Analyze the dynamics near \(x=0\) and show that a bifurcation occurs when \(h=a\). What type of bifurcation is it? e) Show that another bifurcation occurs when \(h=\frac{1}{4}(a+1)^{2}\), for \(a

Ahlers (1989) gives a fascinating review of experiments on one-dimensional patterns in fluid systems. In many cases, the patterns first emerge via supercritical or subcritical pitchfork bifurcations from a spatially uniform state. Near the bifurcation, the dynamics of the amplitude of the patterns are given approximately by \(\tau \dot{A}=\varepsilon A-g A^{3}\) in the supercritical case, or \(\tau \dot{A}=\varepsilon A-g A^{3}-k A^{5}\) in the subcritical case. Here \(A(t)\) is the amplitude, \(\tau\) is a typical time scale, and \(\varepsilon\) is a small dimensionless parameter that measures the distance from the bifurcation. The parameter \(g>0\) in the supercritical case, whereas \(g<0\) and \(k>0\) in the subcritical case. (In this context, the equation \(\tau \dot{A}=\varepsilon A-g A^{3}\) is often called the Landau equation.) a) Dubois and Berg茅 (1978) studied the supercritical bifurcation that arises in Rayleigh-B茅nard convection, and showed experimentally that the steady-state amplitude depended on \(\varepsilon\) according to the power law \(A^{*} \propto \varepsilon^{\beta}\), where \(\beta=0.50 \pm 0.01 .\) What does the Landau equation predict? b) The equation \(\tau \dot{A}=\varepsilon A-g A^{3}-k A^{5}\) is said to undergo a tricritical bifurcation when \(g=0 ;\) this case is the borderline between supercritical and subcritical bifurcations. Find the relation between \(A^{*}\) and \(\varepsilon\) when \(g=0\). c) In experiments on Taylor-Couette vortex flow, Aitta et al, (1985) were able to change the parameter \(g\) continuousiy from positive to negative by varying the aspect ratio of their experimental set-up. Assuming that the equation is modified to \(\tau \dot{A}=h+\varepsilon A-g A^{3}-k A^{5}\), where \(h>0\) is a slight imperfection, sketch the bifurcation diagram of \(A^{*}\) vs. \(\varepsilon\) in the three cases \(g>0, g=0\), and \(g<0 .\) Then look up the actual data in Aitta et al. (1985, Figure 2 ) or see Ahlers \((1989\), Figure 15\()\) d) In the experiments of part (c), the amplitude \(A(t)\) was found to evolve toward a steady state in the manner shown in Figure 2 (redrawn from Ahlers (1989), Figure 18 ). The results are for the imperfect subcritical case \(g<0, h \neq 0\). In the experiments, the parameter \(\varepsilon\) was switched at \(t=0\) from a negative value to a positive value \(\varepsilon_{f} .\) In Figure \(2, \varepsilon_{f}\) increases from the bottom to the top. Explain intuitively why the curves have this strange shape. Why do the curves for large \(\varepsilon_{/}\)go almost straight up to their steady state, whereas the curves for small \(\varepsilon_{f}\) rise to a plateau before increasing sharply to their final level? (Hint: Graph \(\dot{A}\) vs. \(A\) for different \(\varepsilon_{f} .\) )

With tongue in cheek, we pointed out that the pitchfork bifurcation could be called a "trifurcation," since three branches of fixed points appear for \(r>0 .\) Can you construct an example of a "quadfurcation," in which \(\dot{x}=f(x, r)\) has no fixed points for \(r<0\) and four branches of fixed points for \(r>0\) ? Extend your results to the case of an arbitrary number of branches, if possible.

In the following exercises, sketch all the qualitatively different vector fields that occur as \(r\) is varied. Show that a pitchfork bifurcation occurs at a critical value of \(r\) (to be determined) and classify the bifurcation as supercritical or subcritical. Finally, sketch the bifurcation diagram of \(x^{*}\) vs. \(r\). $$ \dot{x}=r x-4 x^{3} $$
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