3-section - 1 3. Microcanonical Ensembles and the Boltzmann...

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1 3. Microcanonical Ensembles and the Boltzmann Distribution 3.0 Introduction Up to now our discussion has focussed on ‘closed’ systems consisting of a Fxed num- ber N of objects, and we have examined the distribution functions associated with arranging those particles in various way (placing them in boxes). Now we extend our discussion to ensembles of “ microcanonical systems ”. A microcanonical system is one which is “ closed ”, which means that no particles may enter or leave, and “ adiabatic ”, or perfectly thermally insulated, which means that no energy may enter or leave in any manner (neither as heat or work), and whose internal energy levels are Fxed and immutable. A microcanonical ensemble is an ensemble consisting of a very large number N of “ equivalent ” microcanonical systems, where equivalent means that all have the same composition (i.e., the same numbers of particles of each type), the same volume V , the same total energy E , and the same set of internal energy levels. ±or a microcanonical molecular system, specifying the composition, represented by the number of particles N , and specifying V e²ectively determines the energy levels for motion of the parti- cles, uniquely specifying the set of all possible energy states of the system { E i } and the associated degeneracies { g ( E i ) } .
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2 e.g., Consider a quantum mechanical system consisting of a single particle of mass m in a cubic box with side length L . E i = E ( n x ,n y z )= h 2 8 mL 2 ± n x 2 + n y 2 + n z 2 ² What are the possible level energies and degeneracies ? n x n y n x E level ig i 111 112 121 311 221 212 122 131 222 321 312 132 . . . . . .
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3 If the particle also has internal energy, must also take account of those contributions to the total energies E i and total degeneracies g i e.g., if the particle is a linear rigid rotor: The for that particle the total energy and degeneracy are If the system contains several particles, the total energy is a sum of terms of this type, one sum for each particle, and the total degeneracy will involve: the product of a degeneracy terms for each particle a sum of those products over all possible ways of arranging the speciFed total energy among those particles At this point it is appropriate to recall a couple of deFnitions introduced earlier: A microstate of a system is deFned by a particular, unique way of specifying the state of the system. A distribution (or confguration ) of a system is a condition of that system charac- terized by a particular set of occupation numbers { n i } or populations for the possible energy levels of the system The Fundamental Postulate of Statistical Mechanics is the principle of equal ´ apr ior i probability.
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3-section - 1 3. Microcanonical Ensembles and the Boltzmann...

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