Gibbs distribution

The distribution of the probabilities of finding a statistical system at equilibrium in any one of its stationary microscopic states. Such states are usually specified as pure quantum-mechanical states defined by a solution $ \psi _ {n} $ of the stationary Schrödinger equation

$$ \widehat{H} \psi _ {n} ( x) = E _ {n} \psi _ {n} ( x), $$

where $ n $ is a complete collection of the quantum numbers defining each such state. A so-called mixed quantum-mechanical state is fully determined by assigning to each state $ \psi _ {n} $ the probability $ w _ {n} $ of the system being found in this state (in case of a continuous spectrum of the quantity $ n $ this will be a probability density). The observed magnitudes are averages over the distribution $ w _ {n} $ of the quantum-mechanical expectation values for each pure state $ n $. The mixed state is fully determined by the statistical von Neumann operator (density matrix), which has the following representation in position space:

$$ ( x | \rho | x ^ \prime ) = \ \sum _ { n } w _ {n} \psi _ {n} ^ {*} ( x ^ \prime ) \psi _ {n} ( x). $$

The observed averages are defined as

$$ \langle F\rangle = \ \sum _ { n } w _ {n} ( \psi _ {n} ^ {*} , \widehat{F} \psi _ {n} ) = \ \mathop{\rm Sp} {\widehat \rho } \widehat{F} . $$

In the case of a Gibbs distribution the mixed state corresponds to the thermodynamic equilibrium state of the system. Since the structure of a Gibbs distribution is $ w _ {n} = w( E _ {n} , A) $, where $ A $ is the totality of thermodynamic parameters that determine the microscopical state of the system, the corresponding operators $ \widehat \rho $ can be directly expressed in terms of the Hamilton operator: $ \widehat \rho = w ( \widehat{H} , A) $, $ \mathop{\rm Sp} \widehat \rho = 1 $. Depending on the choice of the parameters $ A $, different forms of the Gibbs distribution are possible, of which the more frequent ones will be described below.

The microcanonical Gibbs distribution. The parameters $ A $ characterize the state of an isolated system and comprise the energy $ {\mathcal E} $, the volume $ V $, the external fields $ a $, and the number of particles $ N $( in the case of a multi-component system the set of numbers $ N _ {i} $). In such a case the Gibbs distribution takes the form

$$ w _ {n} ( {\mathcal E} , V, a, N) = \ \frac{\Delta ( {\mathcal E} - E _ {n} ) }{\Gamma ( {\mathcal E} , V, a, N) } . $$

Here $ \Gamma $ is the statistical weight, which defines the normalization of the distribution, and which is equal to

$$ \Gamma ( {\mathcal E} , V, a, N) = \ \sum _ { n } \Delta ( {\mathcal E} - E _ {n} ). $$

The sum (or the integral) is taken over all different states of the system, irrespective of their degeneration with respect to $ E _ {n} $. The function $ \Delta ( {\mathcal E} - E _ {n} ) $ is equal to one if the value of $ E _ {n} $ is comprised in an energetic layer $ \delta {\mathcal E} $ around the value of $ {\mathcal E} $ and is zero otherwise. The width $ \delta {\mathcal E} $ should be much smaller than the macroscopic infinitesimal changes in energy $ d {\mathcal E} $, but not smaller than the interval between the energy levels $ \Delta E _ {n} $. The statistical weight $ \Gamma $ determines the number of microscopic states by which a given macroscopic state may be realized, which are all considered as equally probable; it is connected with the entropy of the system by the expression

$$ S ( {\mathcal E} , V, a, N) = \mathop{\rm ln} \Gamma ( {\mathcal E} , V, a, N). $$

The canonical Gibbs distribution. The macroscopic state of the system is defined by the temperature $ \theta $ and by the magnitudes $ V $, $ a $, $ N $( a system "in a heat bath" ). It is the most convenient way of defining a thermodynamic state from the point of view of applications. The canonical Gibbs distribution has the form

$$ w _ {n} ( \theta , V, a, N) = \ \frac{e ^ {- E _ {n} / \theta } }{Z} . $$

Here $ Z $ is the partition function (or sum-over-states):

$$ Z ( \theta , V, a, N) = \ \sum _ { n } e ^ {- E _ {n} / \theta } , $$

which is directly related to the free energy of the system by

$$ F ( \theta , V, a, N) = - \theta \mathop{\rm ln} Z. $$

The grand Gibbs distribution. The parameters $ A $ define the state of the system in a heat bath surrounded by imaginary walls which are freely permeable to the particles. These parameters are $ \theta $, $ V $, $ a $, and the chemical potential $ \mu $( in the case of a multi-component system, several chemical potentials). The Gibbs distribution of microscopic states, as defined by the number of particles $ N $ and by the quantum numbers $ n = n( N) $ of the system of $ N $ bodies, is

$$ w _ {Nn} ( \theta , V, a, \mu ) = \ \frac{e ^ {-( E _ {n} - \mu N)/ \theta } }{Z _ {g} } . $$

Here $ Z _ {g} $ is the grand partition function:

$$ Z _ {g} ( \theta , V, a, \mu ) = \ \sum _ { Nn } e ^ {-( E _ {n} - \mu N)/ \theta } = $$

$$ = \ \sum _ {N = 0 } ^ \infty e ^ {\mu N/ \theta } Z ( \theta , V, a, N), $$

which determines the normalization of this distribution, and is connected with the thermodynamic potential $ \Omega = - pV $, where $ p $ is the pressure, by

$$ \Omega ( \theta , V, a, \mu ) = - \theta \mathop{\rm ln} Z _ {g} . $$

The use of some given Gibbs distribution makes it possible, starting from a given microscopic statistical state of the system, to compute the system's characteristic macroscopic averages, dispersions, etc., and, using the normalized sums $ \Gamma $, $ Z $ or $ Z _ {g} $, to determine all the thermodynamic characteristics of the system at equilibrium. For the choice of a given Gibbs distribution one is led by considerations of convenience. In the statistical limit $ N \rightarrow \infty $, $ V/N = \textrm{ const } $, the results obtained by all Gibbs distributions (expressed in terms of the same variables) are identical. Since the Gibbs method is only valid in this limit, all Gibbs distributions are identical. Microcanonical Gibbs distributions are mainly employed to the general problems of statistical mechanics (the parameters $ A $ do not comprise specific thermodynamic magnitudes of the type $ \theta $, $ \mu $, etc.; canonical Gibbs distributions are mainly employed to classical systems; and grand canonical Gibbs distributions are employed in the studies of quantum systems when, for technical reasons, it is inconvenient to give the exact number $ N $.

At certain values of the parameters $ A $, usually connected with increase in $ \theta $( other parameters remaining unchanged) above the temperature of degeneration (which has a different value for each microscopic state), the general Gibbs distributions become quasi-classical (with respect to the variables for which the motion related to their variation is non-degenerate). In the case of a non-degenerate system of $ N $ particles, when the microscopic motion is represented as the classical motion of $ N $ material points, the microscopic state is defined by a phase point $ n = ( q, p) = ( \mathbf r _ {1} \dots \mathbf r _ {N} , \mathbf p _ {1} \dots \mathbf p _ {N} ) $, the energy is defined by the classical Hamiltonian $ H = H ( q, p) $, while the canonical Gibbs distribution has the form

$$ w _ {qp} ( \theta , V, a, N) dp dq = \ \frac{e ^ {- H ( q, p)/ \theta } }{Z } \frac{dp dq }{N! ( 2 \pi \hbar ) ^ {3N} } . $$

where the classical partition function (quasi-classical limit of the sum-over-states) is

$$ Z = \frac{1}{N! } \int\limits \frac{e ^ {- H ( p, q)/ \theta } }{( 2 \pi \hbar ) ^ {3N} } dp dq. $$

Gibbs distributions were introduced in 1902 by J.W. Gibbs.

References

[1]	J.W. Gibbs, "Elementary principles in statistical mechanics" , Dover, reprint (1960)
[2]	K. Huang, "Statistical mechanics" , Wiley (1963)
[3]	T.L. Hill, "Statistical mechanics" , McGraw-Hill (1956)

Comments

The phrase "Gibbs distribution" is most frequently used to denote the canonical distribution.

References

[a1]	D. Ruelle, "Statistical mechanics: rigorous results" , Benjamin (1974)
[a2]	L.D. Landau, E.M. Lifshitz, "Statistical physics" , A course of theoretical physics , 5 , Pergamon (1969) (Translated from Russian)