In quantum statistics, Bose–
Einstein
Contents 1 Concept
2 History
3 Two derivations of the Bose–
Einstein
3.1 Derivation from the grand canonical ensemble 3.2 Derivation in the canonical approach 4 Interdisciplinary applications 5 See also 6 Notes 7 References Concept[edit]
At low temperatures, bosons behave differently from fermions (which
obey the Fermi–Dirac statistics) in a way that an unlimited number
of them can "condense" into the same energy state. This apparently
unusual property also gives rise to the special state of matter –
the Bose–
Einstein
N V ≥ n q , displaystyle frac N V geq n_ q , where N is the number of particles, V is the volume, and nq is the
quantum concentration, for which the interparticle distance is equal
to the thermal de Broglie wavelength, so that the wavefunctions of the
particles are barely overlapping.
Fermi–Dirac statistics
n i ( ε i ) = g i e ( ε i − μ ) / k T − 1 , displaystyle n_ i (varepsilon _ i )= frac g_ i e^ (varepsilon _ i mu )/kT 1 , with εi > μ and where ni is the number of particles in state i, gi is the degeneracy of energy level i, εi is the energy of the ith state, μ is the chemical potential, k is the Boltzmann constant, and T is absolute temperature. For comparison, the average number of fermions with energy ϵ i displaystyle epsilon _ i given by Fermi–Dirac particleenergy distribution has a similar form: n ¯ i ( ϵ i ) = g i e ( ϵ i − μ ) / k T + 1 . displaystyle bar n _ i (epsilon _ i )= frac g_ i e^ (epsilon _ i mu )/kT +1 . B–E statistics reduces to the
Rayleigh–Jeans law
k T ≫ ϵ i − μ displaystyle kTgg epsilon _ i mu , namely n i = g i k T ε i − μ . displaystyle n_ i = frac g_ i kT varepsilon _ i mu . History[edit]
While presenting a lecture at the
University of Dhaka
Z = ∑ N = 0 ∞ exp ( N ( μ − ϵ ) / k B T ) = ∑ N = 0 ∞ [ exp ( ( μ − ϵ ) / k B T ) ] N = 1 1 − exp ( ( μ − ϵ ) / k B T ) displaystyle begin aligned mathcal Z &=sum _ N=0 ^ infty exp(N(mu epsilon )/k_ B T)=sum _ N=0 ^ infty [exp((mu epsilon )/k_ B T)]^ N \&= frac 1 1exp((mu epsilon )/k_ B T) end aligned and the average particle number for that singleparticle substate is given by ⟨ N ⟩ = k B T 1 Z ( ∂ Z ∂ μ ) V , T = 1 exp ( ( ϵ − μ ) / k B T ) − 1 displaystyle langle Nrangle =k_ B T frac 1 mathcal Z left( frac partial mathcal Z partial mu right)_ V,T = frac 1 exp((epsilon mu )/k_ B T)1 This result applies for each singleparticle level and thus forms the
Bose–
Einstein
⟨ ( Δ N ) 2 ⟩ = k B T ( d ⟨ N ⟩ d μ ) V , T = ⟨ N 2 ⟩ − ⟨ N ⟩ 2 displaystyle langle (Delta N)^ 2 rangle =k_ B Tleft( frac dlangle Nrangle dmu right)_ V,T =langle N^ 2 rangle langle Nrangle ^ 2 This level of fluctuation is much larger than for distinguishable
particles, which would instead show
Poisson statistics
⟨ ( Δ N ) 2 ⟩ = ⟨ N ⟩ 2 displaystyle langle (Delta N)^ 2 rangle =langle Nrangle ^ 2 ). This is because the probability distribution for the number of
bosons in a given energy level is a geometric distribution, not a
Poisson distribution.
Derivation in the canonical approach[edit]
It is also possible to derive approximate Bose–
Einstein
Derivation Suppose we have a number of energy levels, labeled by index i displaystyle displaystyle i , each level having energy ε i displaystyle displaystyle varepsilon _ i and containing a total of n i displaystyle displaystyle n_ i particles. Suppose each level contains g i displaystyle displaystyle g_ i distinct sublevels, all of which have the same energy, and which are distinguishable. For example, two particles may have different momenta, in which case they are distinguishable from each other, yet they can still have the same energy. The value of g i displaystyle displaystyle g_ i associated with level i displaystyle displaystyle i is called the "degeneracy" of that energy level. Any number of bosons can occupy the same sublevel. Let w ( n , g ) displaystyle displaystyle w(n,g) be the number of ways of distributing n displaystyle displaystyle n particles among the g displaystyle displaystyle g sublevels of an energy level. There is only one way of distributing n displaystyle displaystyle n particles with one sublevel, therefore w ( n , 1 ) = 1 displaystyle displaystyle w(n,1)=1 . It is easy to see that there are ( n + 1 ) displaystyle displaystyle (n+1) ways of distributing n displaystyle displaystyle n particles in two sublevels which we will write as: w ( n , 2 ) = ( n + 1 ) ! n ! 1 ! . displaystyle w(n,2)= frac (n+1)! n!1! . With a little thought (see Notes below) it can be seen that the number of ways of distributing n displaystyle displaystyle n particles in three sublevels is w ( n , 3 ) = w ( n , 2 ) + w ( n − 1 , 2 ) + ⋯ + w ( 1 , 2 ) + w ( 0 , 2 ) displaystyle w(n,3)=w(n,2)+w(n1,2)+cdots +w(1,2)+w(0,2) so that w ( n , 3 ) = ∑ k = 0 n w ( n − k , 2 ) = ∑ k = 0 n ( n − k + 1 ) ! ( n − k ) ! 1 ! = ( n + 2 ) ! n ! 2 ! displaystyle w(n,3)=sum _ k=0 ^ n w(nk,2)=sum _ k=0 ^ n frac (nk+1)! (nk)!1! = frac (n+2)! n!2! where we have used the following theorem involving binomial coefficients: ∑ k = 0 n ( k + a ) ! k ! a ! = ( n + a + 1 ) ! n ! ( a + 1 ) ! . displaystyle sum _ k=0 ^ n frac (k+a)! k!a! = frac (n+a+1)! n!(a+1)! . Continuing this process, we can see that w ( n , g ) displaystyle displaystyle w(n,g) is just a binomial coefficient (See Notes below) w ( n , g ) = ( n + g − 1 ) ! n ! ( g − 1 ) ! . displaystyle w(n,g)= frac (n+g1)! n!(g1)! . For example, the population numbers for two particles in three sublevels are 200, 110, 101, 020, 011, or 002 for a total of six which equals 4!/(2!2!). The number of ways that a set of occupation numbers n i displaystyle displaystyle n_ i can be realized is the product of the ways that each individual energy level can be populated: W = ∏ i w ( n i , g i ) = ∏ i ( n i + g i − 1 ) ! n i ! ( g i − 1 ) ! ≈ ∏ i ( n i + g i ) ! n i ! ( g i ) ! displaystyle W=prod _ i w(n_ i ,g_ i )=prod _ i frac (n_ i +g_ i 1)! n_ i !(g_ i 1)! approx prod _ i frac (n_ i +g_ i )! n_ i !(g_ i )! where the approximation assumes that n i ≫ 1 displaystyle n_ i gg 1 . Following the same procedure used in deriving the Maxwell–Boltzmann statistics, we wish to find the set of n i displaystyle displaystyle n_ i for which W is maximised, subject to the constraint that there be a fixed total number of particles, and a fixed total energy. The maxima of W displaystyle displaystyle W and ln ( W ) displaystyle displaystyle ln(W) occur at the same value of n i displaystyle displaystyle n_ i and, since it is easier to accomplish mathematically, we will
maximise the latter function instead. We constrain our solution using
Lagrange multipliers
f ( n i ) = ln ( W ) + α ( N − ∑ n i ) + β ( E − ∑ n i ε i ) displaystyle f(n_ i )=ln(W)+alpha (Nsum n_ i )+beta (Esum n_ i varepsilon _ i ) Using the n i ≫ 1 displaystyle n_ i gg 1 approximation and using
Stirling's approximation
( x ! ≈ x x e − x 2 π x ) displaystyle left(x!approx x^ x ,e^ x , sqrt 2pi x right) gives f ( n i ) = ∑ i ( n i + g i ) ln ( n i + g i ) − n i ln ( n i ) + α ( N − ∑ n i ) + β ( E − ∑ n i ε i ) + K . displaystyle f(n_ i )=sum _ i (n_ i +g_ i )ln(n_ i +g_ i )n_ i ln(n_ i )+alpha left(Nsum n_ i right)+beta left(Esum n_ i varepsilon _ i right)+K. Where K is the sum of a number of terms which are not functions of the n i displaystyle n_ i . Taking the derivative with respect to n i displaystyle displaystyle n_ i , and setting the result to zero and solving for n i displaystyle displaystyle n_ i , yields the Bose–
Einstein
n i = g i e α + β ε i − 1 . displaystyle n_ i = frac g_ i e^ alpha +beta varepsilon _ i 1 . By a process similar to that outlined in the Maxwell–Boltzmann statistics article, it can be seen that: d ln W = α d N + β d E displaystyle dln W=alpha ,dN+beta ,dE which, using Boltzmann's famous relationship S = k ln W displaystyle S=k,ln W becomes a statement of the second law of thermodynamics at constant volume, and it follows that β = 1 k T displaystyle beta = frac 1 kT and α = − μ k T displaystyle alpha = frac mu kT where S is the entropy, μ displaystyle mu is the chemical potential, k is
Boltzmann's constant
n i = g i e ( ε i − μ ) / k T − 1 . displaystyle n_ i = frac g_ i e^ (varepsilon _ i mu )/kT 1 . Note that the above formula is sometimes written: n i = g i e ε i / k T / z − 1 , displaystyle n_ i = frac g_ i e^ varepsilon _ i /kT /z1 , where z = exp ( μ / k T ) displaystyle displaystyle z=exp(mu /kT) is the absolute activity, as noted by McQuarrie.[10] Also note that when the particle numbers are not conserved, removing the conservation of particle numbers constraint is equivalent to setting α displaystyle alpha and therefore the chemical potential μ displaystyle mu to zero. This will be the case for photons and massive particles in mutual equilibrium and the resulting distribution will be the Planck distribution. Notes A much simpler way to think of Bose–
Einstein
( g − 1 + n ) ! ( g − 1 ) ! n ! displaystyle frac (g1+n)! (g1)!n! OR
The purpose of these notes is to clarify some aspects of the
derivation of the Bose–
Einstein
n displaystyle displaystyle n dice, with each die taking values in the set 1 , … , g displaystyle displaystyle 1,dots ,g , for g ≥ 1 displaystyle ggeq 1 . The constraints of the game are that the value of a die i displaystyle displaystyle i , denoted by m i displaystyle displaystyle m_ i , has to be greater than or equal to the value of die ( i − 1 ) displaystyle displaystyle (i1) , denoted by m i − 1 displaystyle displaystyle m_ i1 , in the previous throw, i.e., m i ≥ m i − 1 displaystyle m_ i geq m_ i1 . Thus a valid sequence of die throws can be described by an ntuple ( m 1 , m 2 , … , m n ) displaystyle displaystyle (m_ 1 ,m_ 2 ,dots ,m_ n ) , such that m i ≥ m i − 1 displaystyle m_ i geq m_ i1 . Let S ( n , g ) displaystyle displaystyle S(n,g) denote the set of these valid ntuples: S ( n , g ) = ( m 1 , m 2 , … , m n )
m i ≥ m i − 1 , m i ∈ 1 , … , g , ∀ i = 1 , … , n . displaystyle S(n,g)= Big left(m_ 1 ,m_ 2 ,dots ,m_ n right) Big Big . m_ i geq m_ i1 ,m_ i in left 1,dots ,gright ,forall i=1,dots ,n Big . (1) Then the quantity w ( n , g ) displaystyle displaystyle w(n,g) (defined above as the number of ways to distribute n displaystyle displaystyle n particles among the g displaystyle displaystyle g sublevels of an energy level) is the cardinality of S ( n , g ) displaystyle displaystyle S(n,g) , i.e., the number of elements (or valid ntuples) in S ( n , g ) displaystyle displaystyle S(n,g) . Thus the problem of finding an expression for w ( n , g ) displaystyle displaystyle w(n,g) becomes the problem of counting the elements in S ( n , g ) displaystyle displaystyle S(n,g) . Example n = 4, g = 3: S ( 4 , 3 ) = ( 1111 ) , ( 1112 ) , ( 1113 ) ⏟ ( a ) , ( 1122 ) , ( 1123 ) , ( 1133 ) ⏟ ( b ) , ( 1222 ) , ( 1223 ) , ( 1233 ) , ( 1333 ) ⏟ ( c ) , displaystyle S(4,3)=left underbrace (1111),(1112),(1113) _ (a) ,underbrace (1122),(1123),(1133) _ (b) ,underbrace (1222),(1223),(1233),(1333) _ (c) ,right. ( 2222 ) , ( 2223 ) , ( 2233 ) , ( 2333 ) , ( 3333 ) ⏟ ( d ) displaystyle left.underbrace (2222),(2223),(2233),(2333),(3333) _ (d) right w ( 4 , 3 ) = 15 displaystyle displaystyle w(4,3)=15 (there are 15 displaystyle displaystyle 15 elements in S ( 4 , 3 ) displaystyle displaystyle S(4,3) ) Subset ( a ) displaystyle displaystyle (a) is obtained by fixing all indices m i displaystyle displaystyle m_ i to 1 displaystyle displaystyle 1 , except for the last index, m n displaystyle displaystyle m_ n , which is incremented from 1 displaystyle displaystyle 1 to g = 3 displaystyle displaystyle g=3 . Subset ( b ) displaystyle displaystyle (b) is obtained by fixing m 1 = m 2 = 1 displaystyle displaystyle m_ 1 =m_ 2 =1 , and incrementing m 3 displaystyle displaystyle m_ 3 from 2 displaystyle displaystyle 2 to g = 3 displaystyle displaystyle g=3 . Due to the constraint m i ≥ m i − 1 displaystyle displaystyle m_ i geq m_ i1 on the indices in S ( n , g ) displaystyle displaystyle S(n,g) , the index m 4 displaystyle displaystyle m_ 4 must automatically take values in 2 , 3 displaystyle displaystyle left 2,3right . The construction of subsets ( c ) displaystyle displaystyle (c) and ( d ) displaystyle displaystyle (d) follows in the same manner. Each element of S ( 4 , 3 ) displaystyle displaystyle S(4,3) can be thought of as a multiset of cardinality n = 4 displaystyle displaystyle n=4 ; the elements of such multiset are taken from the set 1 , 2 , 3 displaystyle displaystyle left 1,2,3right of cardinality g = 3 displaystyle displaystyle g=3 , and the number of such multisets is the multiset coefficient ⟨ 3 4 ⟩ = ( 3 + 4 − 1 3 − 1 ) = ( 3 + 4 − 1 4 ) = 6 ! 4 ! 2 ! = 15 displaystyle displaystyle leftlangle begin matrix 3\4end matrix rightrangle = 3+41 choose 31 = 3+41 choose 4 = frac 6! 4!2! =15 More generally, each element of S ( n , g ) displaystyle displaystyle S(n,g) is a multiset of cardinality n displaystyle displaystyle n (number of dice) with elements taken from the set 1 , … , g displaystyle displaystyle left 1,dots ,gright of cardinality g displaystyle displaystyle g (number of possible values of each die), and the number of such multisets, i.e., w ( n , g ) displaystyle displaystyle w(n,g) is the multiset coefficient w ( n , g ) = ⟨ g n ⟩ = ( g + n − 1 g − 1 ) = ( g + n − 1 n ) = ( g + n − 1 ) ! n ! ( g − 1 ) ! displaystyle displaystyle w(n,g)=leftlangle begin matrix g\nend matrix rightrangle = g+n1 choose g1 = g+n1 choose n = frac (g+n1)! n!(g1)! (2) which is exactly the same as the formula for w ( n , g ) displaystyle displaystyle w(n,g) , as derived above with the aid of a theorem involving binomial coefficients, namely ∑ k = 0 n ( k + a ) ! k ! a ! = ( n + a + 1 ) ! n ! ( a + 1 ) ! . displaystyle sum _ k=0 ^ n frac (k+a)! k!a! = frac (n+a+1)! n!(a+1)! . (3) To understand the decomposition w ( n , g ) = ∑ k = 0 n w ( n − k , g − 1 ) = w ( n , g − 1 ) + w ( n − 1 , g − 1 ) + ⋯ + w ( 1 , g − 1 ) + w ( 0 , g − 1 ) displaystyle displaystyle w(n,g)=sum _ k=0 ^ n w(nk,g1)=w(n,g1)+w(n1,g1)+cdots +w(1,g1)+w(0,g1) (4) or for example, n = 4 displaystyle displaystyle n=4 and g = 3 displaystyle displaystyle g=3 w ( 4 , 3 ) = w ( 4 , 2 ) + w ( 3 , 2 ) + w ( 2 , 2 ) + w ( 1 , 2 ) + w ( 0 , 2 ) , displaystyle displaystyle w(4,3)=w(4,2)+w(3,2)+w(2,2)+w(1,2)+w(0,2), let us rearrange the elements of S ( 4 , 3 ) displaystyle displaystyle S(4,3) as follows S ( 4 , 3 ) = ( 1111 ) , ( 1112 ) , ( 1122 ) , ( 1222 ) , ( 2222 ) ⏟ ( α ) , ( 111 3 = ) , ( 112 3 = ) , ( 122 3 = ) , ( 222 3 = ) ⏟ ( β ) , displaystyle S(4,3)=left underbrace (1111),(1112),(1122),(1222),(2222) _ (alpha ) ,underbrace (111 color Red underset = 3 ),(112 color Red underset = 3 ),(122 color Red underset = 3 ),(222 color Red underset = 3 ) _ (beta ) ,right. ( 11 33 == ) , ( 12 33 == ) , ( 22 33 == ) ⏟ ( γ ) , ( 1 333 === ) , ( 2 333 === ) ⏟ ( δ ) ( 3333 ==== ) ⏟ ( ω ) . displaystyle left.underbrace (11 color Red underset == 33 ),(12 color Red underset == 33 ),(22 color Red underset == 33 ) _ (gamma ) ,underbrace (1 color Red underset === 333 ),(2 color Red underset === 333 ) _ (delta ) underbrace ( color Red underset ==== 3333 ) _ (omega ) right . Clearly, the subset ( α ) displaystyle displaystyle (alpha ) of S ( 4 , 3 ) displaystyle displaystyle S(4,3) is the same as the set S ( 4 , 2 ) = ( 1111 ) , ( 1112 ) , ( 1122 ) , ( 1222 ) , ( 2222 ) displaystyle displaystyle S(4,2)=left (1111),(1112),(1122),(1222),(2222)right . By deleting the index m 4 = 3 displaystyle displaystyle m_ 4 =3 (shown in red with double underline) in the subset ( β ) displaystyle displaystyle (beta ) of S ( 4 , 3 ) displaystyle displaystyle S(4,3) , one obtains the set S ( 3 , 2 ) = ( 111 ) , ( 112 ) , ( 122 ) , ( 222 ) displaystyle displaystyle S(3,2)=left (111),(112),(122),(222)right . In other words, there is a onetoone correspondence between the subset ( β ) displaystyle displaystyle (beta ) of S ( 4 , 3 ) displaystyle displaystyle S(4,3) and the set S ( 3 , 2 ) displaystyle displaystyle S(3,2) . We write ( β ) ⟷ S ( 3 , 2 ) displaystyle displaystyle (beta )longleftrightarrow S(3,2) . Similarly, it is easy to see that ( γ ) ⟷ S ( 2 , 2 ) = ( 11 ) , ( 12 ) , ( 22 ) displaystyle displaystyle (gamma )longleftrightarrow S(2,2)=left (11),(12),(22)right ( δ ) ⟷ S ( 1 , 2 ) = ( 1 ) , ( 2 ) displaystyle displaystyle (delta )longleftrightarrow S(1,2)=left (1),(2)right ( ω ) ⟷ S ( 0 , 2 ) = ∅ displaystyle displaystyle (omega )longleftrightarrow S(0,2)=varnothing (empty set). Thus we can write S ( 4 , 3 ) = ⋃ k = 0 4 S ( 4 − k , 2 ) displaystyle displaystyle S(4,3)=bigcup _ k=0 ^ 4 S(4k,2) or more generally, S ( n , g ) = ⋃ k = 0 n S ( n − k , g − 1 ) displaystyle displaystyle S(n,g)=bigcup _ k=0 ^ n S(nk,g1) ; (5) and since the sets S ( i , g − 1 ) , f o r i = 0 , … , n displaystyle displaystyle S(i,g1) , rm for i=0,dots ,n are nonintersecting, we thus have w ( n , g ) = ∑ k = 0 n w ( n − k , g − 1 ) displaystyle displaystyle w(n,g)=sum _ k=0 ^ n w(nk,g1) , (6) with the convention that w ( 0 , g ) = 1 , ∀ g , a n d w ( n , 0 ) = 1 , ∀ n displaystyle displaystyle w(0,g)=1 ,forall g , rm and w(n,0)=1 ,forall n . (7) Continuing the process, we arrive at the following formula w ( n , g ) = ∑ k 1 = 0 n ∑ k 2 = 0 n − k 1 w ( n − k 1 − k 2 , g − 2 ) = ∑ k 1 = 0 n ∑ k 2 = 0 n − k 1 ⋯ ∑ k g = 0 n − ∑ j = 1 g − 1 k j w ( n − ∑ i = 1 g k i , 0 ) . displaystyle displaystyle w(n,g)=sum _ k_ 1 =0 ^ n sum _ k_ 2 =0 ^ nk_ 1 w(nk_ 1 k_ 2 ,g2)=sum _ k_ 1 =0 ^ n sum _ k_ 2 =0 ^ nk_ 1 cdots sum _ k_ g =0 ^ nsum _ j=1 ^ g1 k_ j w(nsum _ i=1 ^ g k_ i ,0). Using the convention (7)2 above, we obtain the formula w ( n , g ) = ∑ k 1 = 0 n ∑ k 2 = 0 n − k 1 ⋯ ∑ k g = 0 n − ∑ j = 1 g − 1 k j 1 , displaystyle displaystyle w(n,g)=sum _ k_ 1 =0 ^ n sum _ k_ 2 =0 ^ nk_ 1 cdots sum _ k_ g =0 ^ nsum _ j=1 ^ g1 k_ j 1, (8) keeping in mind that for q displaystyle displaystyle q and p displaystyle displaystyle p being constants, we have ∑ k = 0 q p = q p displaystyle displaystyle sum _ k=0 ^ q p=qp . (9) It can then be verified that (8) and (2) give the same result for w ( 4 , 3 ) displaystyle displaystyle w(4,3) , w ( 3 , 3 ) displaystyle displaystyle w(3,3) , w ( 3 , 2 ) displaystyle displaystyle w(3,2) , etc. Interdisciplinary applications[edit] Viewed as a pure probability distribution, the Bose–Einstein distribution has found application in other fields: In recent years, Bose
Einstein
Main article: Bose–
Einstein
See also[edit] Bose–
Einstein
Notes[edit] ^ See p. 14, note 3, of the Ph.D. Thesis entitled Bose–Einstein
condensation: analysis of problems and rigorous results, presented by
Alessandro Michelangeli to the International School for Advanced
Studies, Mathematical Physics Sector, October 2007 for the degree of
Ph.D. See: "Archived copy". Archived from the original on 20131106.
Retrieved 20120325. ?show=full, and download from "Archived
copy". Archived from the original on 20131106. Retrieved
20120325.
^ Bose (2 July 1924). "
Planck's law
References[edit] Annett, James F. (2004). Superconductivity, Superfluids and Condensates. New York: Oxford University Press. ISBN 0198507550. Carter, Ashley H. (2001). Classical and Statistical Thermodynamics. Upper Saddle River, New Jersey: Prentice Hall. ISBN 0137792085. Griffiths, David J. (2005). Introduction to Quantum Mechanics (2nd ed.). Upper Saddle River, New Jersey: Pearson, Prentice Hall. ISBN 0131911759. McQuarrie, Donald A. (2000). Statistical Mechanics (1st ed.). Sausalito, California 94965: University Science Books. p. 55. ISBN 1891389157. v t e Statistical mechanics Statistical ensembles Microcanonical Canonical Grand canonical Isothermal–isobaric Isoenthalpic–isobaric ensemble Statistical thermodynamics Characteristic state functions Partition functions Translational Vibrational Rotational Equations of state Dieterici Van der Waals/Real gas law Ideal gas law Birch–Murnaghan Entropy Sackur–Tetrode equation Tsallis entropy Von Neumann entropy Particle statistics Maxwell–Boltzmann statistics
Fermi–Dirac statistics
Bose–
Einstein
Statistical field theory Conformal field theory Osterwalder–Schrader axioms Quantum statistical mechanics Density matrix
Gibbs measure
Partition function
Phase space formulation
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