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11 Energy11.1 Magnetic Dipole Model11.2 Principle of Maximum Entropy for Physical Systems11.2.1 General Properties11.2.2 Differential Forms11.2.3 Differential Forms with External Parameters11.3 System and Environment11.3.1 Partition Model11.3.2 Interaction Model11.3.3 Extensive and Intensive Quantities11.3.4 Equilibrium11.3.5 Energy Flow, Work and Heat11.3.6 Reversible Energy FlowChapter 11EnergyIn Chapter 9 of these notes we introduced the Principle of Maximum Entropy as a technique for estimatingprobability distributions consistent with constraints.A simple case that can be done analytically is that in which there are three probabilities, one constraint inthe form of an average value, and the fact that the probabilities add up to one. There are, then, two equationsand three unknowns, and it is straightforward to express the entropy in terms of one of the unknowns,eliminate the others, and find the maximum. This approach also works if there are four probabilities andtwo average-value constraints, in which case there is again one fewer equation than unknown.Another special case is one in which there are many probabilities but only one average constraint. Al-though the e ntropy cannot be expressed in terms of a single probability, the solution in Chapter 9 is practicalif the summations can be calculated.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 bec ause it leads to relationships among differentquantities. In this chapter we look at general features of such systems.Because we are now interested in physical systems, we will express the entropy in Joules per Kelvin ratherthan in bits, and use the natural logarithm rather than the logarithm to the base 2.11.1 Magnetic Dipole ModelMost of the results below apply to the general multi-state model of a physical system implied by quan-tum mechanics, from Chapter 10. However, an imp ortant aspect is the dependence of energy on externalparameters. For example, for the magnetic dipole, the external parameter is the magnetic field H. Here isa brief review of the magnetic dipole so it can be used as an example below.This model was introduced in section 9.1.2. Figure 11.1 shows a system with two dipoles. (Of course anypractical system will have many more than two dipoles, but the important ideas are illustrated with onlytwo.) The dipoles are subjected to an externally applied magnetic field H, and therefore the energy of thesystem depends on the orientations of the dipoles and on the applied field. Our system, with exactly twosuch dipoles, will be able to interchange information and energy with either of two environments, which aremuch larger collections of similar dipoles. Each dipole, both in the system and in its two environments, canbe either “up” or “down,” so there are four possible states for the sys tem, “up-up,” “up-down,” “down-up,”and “down-down.” The energy of a dipole is mdH if down and −mdH if up, and the energy of each of thefour states is the sum of the energies of the two dipoles.Author: Paul Penfield, Jr.This document: ht tp:/ /www -mtl .mit .ed u/Co urse s/6. 050/ 200 5/no tes/ chap ter1 1.p dfVersion 1.2, April 25, 2005. Copyrightc 2005 Massachusetts Institute of TechnologyStart of notes · back · next | 6.050J/2.110J home page | Site map | Search | Ab out this document | Comments and inquiries11111.2 Principle of Maximum Entropy for Physical Systems 112↑ H ↑⊗ ⊗ ⊗· · ·⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗· · ·⊗ ⊗ ⊗Figure 11.1: Dipole moment example.(Each dipole can be either up or down.)11.2 Principle of Maximum Entropy for Physical SystemsAccording to the multi-state model motivated by quantum mechanics (see Chapter 10 of these notes)there are a finite (or countable infinite) number of quantum states of the system. We will use i as an indexover these states. The states have energy Ei, and might have other physical attributes as well. After thesestates are enumerated and describ ed, then the Principle of Maximum Entropy can be used, as a separatestep, to estimate 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 being a known (for example,measured) quantityeE. ThuseE =XipiEi(11.1)1 =Xipi(11.2)The entropy isS = kBXipiln1pi(11.3)where kB= 1.38 × 10−23Joules per Kelvin and is known as Boltzmann’s constant.The probability distribution that maximizes S subject to a constraint like Equation 11.2 was presentedin Chapter 9, Equation 9.12. That formula was for the case where entropy was expressed in bits; thecorresponding formula for physical systems, with entropy expressed in Joules per Kelvin, is the same exceptfor the use of e rather than 2:pi= e−αe−βEi(11.4)so thatln1pi= α + βEi(11.5)The sum of the probabilities must be 1 and thereforeα = ln Xie−βEi!(11.6)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 easier 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.11.2 Principle of Maximum Entropy for Physical Systems 11311.2.1 General PropertiesBecause β plays a central role, it is helpful to understand intuitively how different values it may assumeaffect things.First, if β = 0, all probabilities are equal. T his can only happen when there are a finite number of states.Second, if β > 0, then states with lower energy have a higher probability of being occupied. Similarly, ifβ < 0, then states with higher energy have a higher probability of being occupied. Because of the exponentialdependence on energy, unless | β | is small, the only states with much probability of being occupied are thosewith energy close to the maximum possible (β negative) or minimum possible (β positive).Third, we can multiply the equation above for ln(1/pi) by piand sum over i to obtainS = kB(α + βE) (11.7)This equation is


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