10.8 The Partition Function for Ensembles
455
10.8.1 Gibbs’ Ensemble
In 1902, Willard Gibbs published his analysis over many years 30 on the statistics
of systems near thermodynamic equilibrium, establishing the subject of statistical
mechanics on a firm theoretical basis. 31 Gibbs generalized Boltzmann’s ideas to
any macroscopic systems whose energies for all the contained subsystems are
known, including interaction energies. This is in contrast to our previous discussion
applying statistics to systems whose individual particles act independently.
To describe the statistics of large systems, Gibbs imagined N replicas of a
system, each copy initially having the same complete set of macroscopic variables,
such as composition, energy, and volume. Also, all possible microstates consistent
with the macroscopic variables are equally represented in the initial collection,
before each system in the ensemble is allowed to come to equilibrium. To insure
a probabilistic interpretation for the chance of any particular state occurring, the
number of systems in the ensemble, N , is imagined taken to infinity. Such a
collection is called a ‘Gibbs ensemble’.
10.8.2 Systems Described by Classical Theory
Classically, the microscopic dynamical laws of a system of N particles can be
expressed in terms of the time behavior of the full set of coordinates q ≡ {q i ; i =
1, 3N } and their ‘conjugate’ momenta p ≡ {p i ; i = 1, 3N } for the contained
particles. Generalized coordinates and conjugate momenta arise in the Lagrange
formulation of mechanics, wherein Newtonian dynamics is determined by the
minimization the ‘action’ S A =
t 2
t 1
Ldt, a time integral over a function L of the
coordinates q and velocities ˙
q of the particles. The function L(q, ˙
q, t) is called
the ‘Lagrangian’. The minimum of the action is found by varying the possible
paths q
i (t) = q i (t) + δq(t) (holding the initial and final values unchanged).
Newton’s laws result when this variation of the action vanishes. They take the
form (d/dt)∂L/∂ ˙
q i = ∂L/∂q i . The momentum conjugate to q i is defined to be
p i ≡ ∂L/∂ ˙
q i . The energy of the system, expressed in terms of the q and p, is
called the ‘Hamiltonian’, which turns out to be H =
i (∂L/∂ ˙
q) ˙
q − L. With the
Hamiltonian, Newton’s laws come from the pair of equations dq i /dt = ∂H/∂p i
and dp i /dt = −∂H/∂q i .
Therefore, classically, the time-evolution of each member of a Gibbs ensemble
is determined by the motion of a point in the space of all 6N variables {q i , p i },
30 J. Willard Gibbs, Elementary Principles in Statistical Mechanics, Developed with especial
reference to the rational foundations of thermodynamics, Yale University, [Scribners Sons]
(1902).
31 Gibbs’ work was praised by leading physicists of the time for its logic and depth, and is still
applicable even after the advent of quantum theory.
455
10.8.1 Gibbs’ Ensemble
In 1902, Willard Gibbs published his analysis over many years 30 on the statistics
of systems near thermodynamic equilibrium, establishing the subject of statistical
mechanics on a firm theoretical basis. 31 Gibbs generalized Boltzmann’s ideas to
any macroscopic systems whose energies for all the contained subsystems are
known, including interaction energies. This is in contrast to our previous discussion
applying statistics to systems whose individual particles act independently.
To describe the statistics of large systems, Gibbs imagined N replicas of a
system, each copy initially having the same complete set of macroscopic variables,
such as composition, energy, and volume. Also, all possible microstates consistent
with the macroscopic variables are equally represented in the initial collection,
before each system in the ensemble is allowed to come to equilibrium. To insure
a probabilistic interpretation for the chance of any particular state occurring, the
number of systems in the ensemble, N , is imagined taken to infinity. Such a
collection is called a ‘Gibbs ensemble’.
10.8.2 Systems Described by Classical Theory
Classically, the microscopic dynamical laws of a system of N particles can be
expressed in terms of the time behavior of the full set of coordinates q ≡ {q i ; i =
1, 3N } and their ‘conjugate’ momenta p ≡ {p i ; i = 1, 3N } for the contained
particles. Generalized coordinates and conjugate momenta arise in the Lagrange
formulation of mechanics, wherein Newtonian dynamics is determined by the
minimization the ‘action’ S A =
t 2
t 1
Ldt, a time integral over a function L of the
coordinates q and velocities ˙
q of the particles. The function L(q, ˙
q, t) is called
the ‘Lagrangian’. The minimum of the action is found by varying the possible
paths q
i (t) = q i (t) + δq(t) (holding the initial and final values unchanged).
Newton’s laws result when this variation of the action vanishes. They take the
form (d/dt)∂L/∂ ˙
q i = ∂L/∂q i . The momentum conjugate to q i is defined to be
p i ≡ ∂L/∂ ˙
q i . The energy of the system, expressed in terms of the q and p, is
called the ‘Hamiltonian’, which turns out to be H =
i (∂L/∂ ˙
q) ˙
q − L. With the
Hamiltonian, Newton’s laws come from the pair of equations dq i /dt = ∂H/∂p i
and dp i /dt = −∂H/∂q i .
Therefore, classically, the time-evolution of each member of a Gibbs ensemble
is determined by the motion of a point in the space of all 6N variables {q i , p i },
30 J. Willard Gibbs, Elementary Principles in Statistical Mechanics, Developed with especial
reference to the rational foundations of thermodynamics, Yale University, [Scribners Sons]
(1902).
31 Gibbs’ work was praised by leading physicists of the time for its logic and depth, and is still
applicable even after the advent of quantum theory.