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3 Ensembles: Systems of Particles
completely precise description of an isolated system of particles would need to
take into account all interactions amongst its particles, and would determine the
rigorously exact quantum states of the system. In fact, were the system found in
any one of these exact states, it would then remain in that state forever. In practice,
no system is ever so isolated from the rest of the environment for these conditions
to be obtained, and we never know the exact microscopic states of a system of
particles in interaction. Thus, we employ approximate states determined by taking
into account all the important dynamical properties of its particles, while neglecting
small residual interactions.
A system initially known to be in one of its approximate quantum states does
not remain there forever. In the course of time, under the influence of small residual
interactions, it will make a transition from that state into another state (provided
that there is no violation of one of the laws of mechanics). Such states are called
accessible states of the system. One of our objectives is to be able to determine
those states that are accessible to a given system.
The typical specifications for a system of particles are energy E, pressure P ,
and so on. Let us call the number of states accessible to the system, and let
{y 1 , y 2 , . . . , y n } represent the set of all constraints imposed upon the system. Clearly
= 1 , y 2 , . . . , y n ), which implies that the number of accessible states is
a function of the parameters y 1 , y 2 , . . . , y n . The most commonly occurring such
parameter is y = E. We shall therefore seek an expression for the number of
accessible states with energies lying between E and E + δE, with δE E.
Of course in addition to we are also interested in knowing in which one
of the accessible states will the system be found. Is this an answerable question?
In complete analogy with the tossing of a die, we cannot know in which state our
system lies, but we can calculate the probability that the system lies in any one of its
accessible states. If, for example, there are accessible states, then it is extremely
difficult to see any reason why the system should be preferably in any specific one
of these states, and so we guess that the probability will be p = 1// for any
given state. This is known as the Fundamental Postulate of Statistical Mechanics
or, equivalently, as the Equal a priori Probability Postulate.
Fundamental Postulate of Statistical Mechanics. If an isolated system is in equilibrium,
it is found with equal probability in each one of its accessible states.
If we combine the Fundamental Postulate with the postulate of long-time averages [stated in the introductory chapter (see Sect. 1.4)], we are led to the conclusion
that a single isolated system of a microcanonical ensemble, considered over a very
long time period, spends equal amounts of time in each of its accessible quantum
states. This result is commonly referred to as the quantum ergodic hypothesis. To
broaden its application to the canonical and grand ensembles, it suffices to note that
these ensembles themselves are isolated systems.
3 Ensembles: Systems of Particles
completely precise description of an isolated system of particles would need to
take into account all interactions amongst its particles, and would determine the
rigorously exact quantum states of the system. In fact, were the system found in
any one of these exact states, it would then remain in that state forever. In practice,
no system is ever so isolated from the rest of the environment for these conditions
to be obtained, and we never know the exact microscopic states of a system of
particles in interaction. Thus, we employ approximate states determined by taking
into account all the important dynamical properties of its particles, while neglecting
small residual interactions.
A system initially known to be in one of its approximate quantum states does
not remain there forever. In the course of time, under the influence of small residual
interactions, it will make a transition from that state into another state (provided
that there is no violation of one of the laws of mechanics). Such states are called
accessible states of the system. One of our objectives is to be able to determine
those states that are accessible to a given system.
The typical specifications for a system of particles are energy E, pressure P ,
and so on. Let us call the number of states accessible to the system, and let
{y 1 , y 2 , . . . , y n } represent the set of all constraints imposed upon the system. Clearly
= 1 , y 2 , . . . , y n ), which implies that the number of accessible states is
a function of the parameters y 1 , y 2 , . . . , y n . The most commonly occurring such
parameter is y = E. We shall therefore seek an expression for the number of
accessible states with energies lying between E and E + δE, with δE E.
Of course in addition to we are also interested in knowing in which one
of the accessible states will the system be found. Is this an answerable question?
In complete analogy with the tossing of a die, we cannot know in which state our
system lies, but we can calculate the probability that the system lies in any one of its
accessible states. If, for example, there are accessible states, then it is extremely
difficult to see any reason why the system should be preferably in any specific one
of these states, and so we guess that the probability will be p = 1// for any
given state. This is known as the Fundamental Postulate of Statistical Mechanics
or, equivalently, as the Equal a priori Probability Postulate.
Fundamental Postulate of Statistical Mechanics. If an isolated system is in equilibrium,
it is found with equal probability in each one of its accessible states.
If we combine the Fundamental Postulate with the postulate of long-time averages [stated in the introductory chapter (see Sect. 1.4)], we are led to the conclusion
that a single isolated system of a microcanonical ensemble, considered over a very
long time period, spends equal amounts of time in each of its accessible quantum
states. This result is commonly referred to as the quantum ergodic hypothesis. To
broaden its application to the canonical and grand ensembles, it suffices to note that
these ensembles themselves are isolated systems.