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MIT 6 050J - Energy

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12 Energy12.1 Magnetic Dipole Model12.2 Principle of Maximum Entropy for Physical Systems12.2.1 General Properties12.3 System and Environment12.3.1 Partition Model12.3.2 Interaction Model12.3.3 Extensive and Intensive Quantities12.3.4 Equilibrium12.3.5 Energy Flow, Work and Heat12.3.6 Reversible Energy FlowChapter 12EnergyIn previous chapters of these notes we introduced the Principle of Maximum Entropy as a technique forestimating probability distributions consistent with constraints.In Chapter 9 we discussed the simple case which can be done analytically, in which there are threeprobabilities, one constraint in the form of an average value, and the fact that the probabilities add up toone. There are, then, two equations and three unknowns, and it is straightforward to express the entropyin terms of one of the unknowns, eliminate the others, and find the maximum. This approach also works ifthere are four probabilities and two average-value constraints, in which case there is again one fewer equationthan unknown.In Chapter 10 we discussed a more general case in which there are many probabilities but only oneaverage constraint, so that the entropy cannot be expressed in terms of a single probability. The method ofLagrange multipliers was used, and provided the summations can be done, a general method of solution waspresented.In the application of the Principle of Maximum Entropy to physical systems, the number of possiblestates is usually very large, so that neither analytic nor numerical solutions are practical. Even in thiscase, however, the Principle of Maximum Entropy is useful because it leads to relationships among differentquantities. Here we look at ge neral features of such systems including the dependence of some quantities onothers. A relatively simple model of a magnetic dipole system will be used.Because we are now interested in the application to physical systems, we will express the entropy inJoules per Kelvin rather than in bits, and use the natural logarithm rather than the logarithm to the base2. We will assume the multi-state model of the physical system implied by quantum mechanics, as discussedin Chapter 11 of these notes. This chapter focuses on the flow of energy in physical s ystems .12.1 Magnetic Dipole ModelThis model was introduced in subsection 10.1.2, and is shown below in Figure 12.1. An array of magneticdipoles (think of them as tiny magnets) is subjected to an externally applied magnetic field H, and thereforethe energy of the system depends on the orientations of the dipoles and on the applied field. Our systemcontains exactly two such dipoles, but it will from time to time be able to interchange information andenergy with either of two environments, which are much larger collections of dipoles. Each dipole, both inthe system and in its two environments, can be either “up” or “down,” so there are four possible statesfor the system, “up-up,” “up-down,” “down-up,” and “down-down.” The energy of a dipole depends on itsorientation and is proportional to the strength of the applied field, and the energy of each state is the sumAuthor: Paul Penfield, Jr.Version 1.0.2, May 1, 2003. Copyrightc 2003 Massachusetts Institute of TechnologyURL: http://www-mtl.mit.edu/Courses/6.050/notes/chapter12.pdfTOC: http://www-mtl.mit.edu/Courses/6.050/notes/index.html10012.2 Principle of Maximum Entropy for Physical Systems 101↑ H ↑⊗ ⊗ ⊗· · ·⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗· · ·⊗ ⊗ ⊗Figure 12.1: Dipole moment example.(Each dipole can b e e ither up or down.)of the energies of the two dipoles. We will deal with the two-dipole system for simplicity, and in some casesan even simpler model with only one dip ole.12.2 Principle of Maximum E ntropy for Physical SystemsAccording to the multi-state model motivated by quantum mechanics (see Chapter 11 of these notes) thereare a finite (or countable infinite) number of quantum states of the system. We will use i as an index overthese states. The states have energy Ei, and might have other physical attributes as well. After these statesare enumerated and described, then the Principle of Maximum Entropy can be used, as a separate step, toestimate which states are likely to be occupied.We denote the occupancy of state i by the event Ai. The state i has probability p(Ai) of being occupied.For simplicity we will write this probability p(Ai) as pi. We use the Principle of Maximum Entropy toestimate the probability distribution piconsistent with the average energy E b e ing a known (for example,measured) quantity. ThusE =XipiEi(12.1)1 =Xipi(12.2)The entropy isS = kBXipiln1pi(12.3)where kB= 1.38 × 10−23Joules per Kelvin and is known as Boltzmann’s constant.In the technique of Lagrange Multipliers, we do not use these equations to reduce the number of variables,but instead increase the numbe r of unknowns. We define the “Lagrange multipliers” α and β and then the“Lagrangian” function LL = S − kB(α − 1)"Xipi− 1#− kBβ"XiEipi− E#(12.4)The Lagrange multiplier α is dimensionless and β is expressed in inverse Joules.The Principle of Maximum Entropy is carried out by finding α, β, and all pisuch that L is made thelargest. These values of pialso make S the largest it can be, subject to the constraints.Since α only appears once in the expression for L, the quantity that multiplies it must be zero for thevalues that maximize L (otherwise a small change in α could increase L). Similarly, β only appears once inthe expression for L, so the quantity that multiplies it must also vanish. Thus in the general case the pithatwe are seeking must satisfy12.2 Principle of Maximum Entropy for Physical Systems 1020 =Xipi− 1 (12.5)0 =XiEipi− E (12.6)The result of maximizing L with respect to each of the piis a relation between piand the Lagrangemultipliersln1pi= α + βEi(12.7)sopi= e−αe−βEi(12.8)The sum of the probabilities must be 1 and thereforeα = ln Xie−βEi!(12.9)As expressed in terms of the Principle of Maximum Entropy, the objective is to find the various quantitiesgiven the expected energy E. However, except in the simplest circumstances it is usually eas ier to docalculations the other way around. That is, it is easier to use β as an independent variable, calculate α interms of it, and then find the piand then the entropy S and energy E.Because the Lagrange multiplier plays a central role, it is helpful to understand intuitively how


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MIT 6 050J - Energy

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