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CH06_ERGODIC

# CH06_ERGODIC - Chapter 6 Ergodicity and the Approach to...

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Chapter 6 Ergodicity and the Approach to Equilibrium 6.1 Equilibrium Recall that a thermodynamic system is one containing an enormously large number of constituent particles, a typical ‘large number’ being Avogadro’s number, N A = 6 . 02 × 10 23 . Nevertheless, in equilibrium , such a system is characterized by a relatively small number of thermodynamic state variables. Thus, while a complete description of a (classical) system would require us to account for O ( 10 23 ) evolving degrees of freedom, with respect to the physical quantities in which we are interested, the details of the initial conditions are effectively forgotten over some microscopic time scale τ , called the collision time, and over some microscopic distance scale, , called the mean free path 1 . The equilibrium state is time-independent. 6.2 The Master Equation Relaxation to equilibrium is often modeled with something called the master equation . Let P i ( t ) be the probability that the system is in a quantum or classical state i at time t . Then write dP i dt = summationdisplay j ( W ji P j W ij P i ) . (6.1) Here, W ij is the rate at which i makes a transition to j . Note that we can write this equation as dP i dt = summationdisplay j Γ ij P j , (6.2) 1 Exceptions involve quantities which are conserved by collisions, such as overall particle number, mo- mentum, and energy. These quantities relax to equilibrium in a special way called hydrodynamics . 1

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2 CHAPTER 6. ERGODICITY AND THE APPROACH TO EQUILIBRIUM where Γ ij = braceleftBigg W ji if i negationslash = j k W jk if i = j , (6.3) where the prime on the sum indicates that k = j is to be excluded. The constraints on the W ij are that W ij 0 for all i,j , and we may take W ii 0 (no sum on i ). Fermi’s Golden Rule of quantum mechanics says that W ji = 2 π planckover2pi1 vextendsingle vextendsingle ( i | ˆ V | j ) vextendsingle vextendsingle 2 ρ ( E i ) , (6.4) where ˆ H 0 vextendsingle vextendsingle i )big = E i vextendsingle vextendsingle i )big , ˆ V is an additional potential which leads to transitions, and ρ ( E i ) is the density of final states at energy E i . If the transition rates W ij are themselves time-independent, then we may formally write P i ( t ) = ( e Γ t ) ij P j (0) . (6.5) Here we have used the Einstein ‘summation convention’ in which repeated indices are summed over (in this case, the j index). Note that summationdisplay i Γ ij = 0 , (6.6) which says that the total probability i P i is conserved: d dt summationdisplay i P i = summationdisplay i,j Γ ij P j = summationdisplay j parenleftBig summationdisplay i Γ ij parenrightBig P j = 0 . (6.7) Suppose we have a time-independent solution to the master equation, P eq i . Then we must have Γ ij P eq j = 0 = P eq j W ji = P eq i W ij . (6.8) This is called the condition of detailed balance . Assuming W ij negationslash = 0 and P eq j = 0, we can divide to obtain W ji W ij = P eq i P eq j . (6.9) 6.2.1 Example: radioactive decay Consider a group of atoms, some of which are in an excited state which can undergo nuclear decay. Let P n ( t ) be the probability that n atoms are excited at some time
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CH06_ERGODIC - Chapter 6 Ergodicity and the Approach to...

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