lect5 - TABLE I Road Map of Ensembles ensemble macrostate...

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Unformatted text preview: TABLE I: Road Map of Ensembles ensemble macrostate probability distribution thermodynamics microcanonical E,V,N p i = 1 / Ω S ( E,V,N ) = k ln(Ω) canonical T,V,N p i = exp( − βE i ) /Z F ( T,V,N ) = − kT ln( Z ) grandcanonical T,V,μ p i = exp( − β ( E i − γN i ) / Z Φ( T,V,μ ) = − k ln( Z ) The Grand Canonical Ensemble POCKET BOOK REFERENCE: GRAND CANONICAL ENSEMBLE Canonical ensemble: System and a (very large) heat bath/reservoir in equilibrium. Fixed T,V,N . The probability of finding the system in a microstate i with energy E i : p ij = e- E i- μN j kT Z (Gibbs distribution) . Canonical partition function: Z ( T,V,μ ) = summationdisplay ij e- E i- μN j kT . Grand potential of the canonical ensemble: Φ( T,V,μ ) = U − TS − μN = − kT ln Z ( T,V,μ ) , GRAND CANONICAL ENSEMBLE Consider a system which is coupled to a heat and particle reservoir, that is it can exchange energy and particles with its environment. Suppose that that the energy and particle number of the system is much smaller than that of the reservoir, N S ≪ N R and E S ≪ E R . Consider an ensemble of N such systems and let n ij is the number of systems that have energy E i and N j particles. Using the Lagrange multiplier method, we can determine the most probable distribution set, n * ij : 1 TABLE II: Ensemble distributions with a given particle number and energy. Number of ensembles Energy Number of particles n 11 E 1 N 1 n 12 E 1 N 2 n 21 E 2 N 1 . . . Each distribution (see the Table) must satisfy the following constrains: summationdisplay ij n ij = N summationdisplay ij E i n ij = N( E ) summationdisplay ij N j n ij = N( N ) There are two indeces since neither the energy not the number of particles are the same in all systems. The probability of the distribution { n ij } is proportional to the number of ways, W { n ij } , the given distribution can be formed. The total number of ways W { n ij } to generate a certain distribution { n ij } is W { n ij } = N ! producttext ij n ij ! . (1) The most probable distribution is the one which maximizes the weight factor W { n ij } . To find the maxima, again, it is simpler to work with ln( W ): ln( W ) = ln( N !) − summationdisplay ij ln( n ij !) (2) We are in the N → ∞ and n i → ∞ regime, so we can use the Stirling formula. ln( W ) = N ln( N ) − summationdisplay ij n ij ln( n ij ) (3) To determine the maxima of ln( W ) subject to the constrains we use the Lagrange multiplier technique. First we form a functional by adding the constrains multiplied by Lagrange multipliers to ln( W ) L [ n ij ,α,β,γ ] = ln( W ) + α ( N − summationdisplay ij n ij ) + β ( N( E )− summationdisplay ij n ij E i ) − γ ( N( N )− summationdisplay ij n ij N j ) To find the minima subject to the constrains one has to require that ∂L ∂n i = 0 , ∂L ∂α = 0 , ∂L ∂β = 0 , ∂L ∂γ = 0 ....
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lect5 - TABLE I Road Map of Ensembles ensemble macrostate...

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