48
L. Rondoni
constant, one obtains that f
(s)
N remains factorized in time, as product of a number s
of f
(1)
N factors, if it is so factorized at the beginning [22].
Given the H-functional defined by:
H =
f log f dp
(1.186)
where f is a 1-particle distribution, one of the main conceptual results stemming
from the Boltzmann equation is the H-theorem. Provided f is the solution of the
Boltzmann equation under boundary conditions of regular relflecting walls, hence
for an isolated system, the theorem states:
d
dt
H(q) dq ≤ 0
(1.187)
where equality only holds when f is the Maxwell-Boltzmann distribution.
For isolated dilute gases, H reduces to the celebrated Boltzmann entropy
S B = k B log W
(1.188)
that states that equal volumes in phase space correspond to events of equal probability, because the number W of different ways in which a given thermodynamc state
can be microscopically realized is identified with a given volume in phase space.
Equation (1.188) is also called bridge law, since it connects the microscopic description afforded by W with the macroscopic thermodynamic description provided by
the entropy and its derivatives, via the amazing Boltzmann constant k B . Also known
as Boltzmann postulate, Eq. (1.188), was rewritten by Einstein as:
Pr(state) ∝ e
S/K B
(1.189)
meaning that a fluctuation away from the equilibrium state, characterized by a (negative) variation of entropy S, is exponentially unlikely. This begins the industry of
fluctuation theories, and imparts momentum to stochastic modeling in Physics.
The distribution f appearng in the Boltzmann equation is a probability distribution
obtained by projecting down the probability distribution for N particles on the 2d N
exceedingly high dimensional phase space, to the 2d dimensional 1-particle space.
17
In other words,
f (q, p; t)dqdp
(1.190)
represents the probability of finding at time t one particle in the elementary volume
dqdp, i.e. to find one particle within a volume of size dq around q, with momentum
within a set of size dp around p. Furthermore, probability is intended here as the
17 The 2d-dimensional space of a single particle positions and velocities is called μ-space, to distinguish it from the 6N dimensional -space, which is the phase space.
L. Rondoni
constant, one obtains that f
(s)
N remains factorized in time, as product of a number s
of f
(1)
N factors, if it is so factorized at the beginning [22].
Given the H-functional defined by:
H =
f log f dp
(1.186)
where f is a 1-particle distribution, one of the main conceptual results stemming
from the Boltzmann equation is the H-theorem. Provided f is the solution of the
Boltzmann equation under boundary conditions of regular relflecting walls, hence
for an isolated system, the theorem states:
d
dt
H(q) dq ≤ 0
(1.187)
where equality only holds when f is the Maxwell-Boltzmann distribution.
For isolated dilute gases, H reduces to the celebrated Boltzmann entropy
S B = k B log W
(1.188)
that states that equal volumes in phase space correspond to events of equal probability, because the number W of different ways in which a given thermodynamc state
can be microscopically realized is identified with a given volume in phase space.
Equation (1.188) is also called bridge law, since it connects the microscopic description afforded by W with the macroscopic thermodynamic description provided by
the entropy and its derivatives, via the amazing Boltzmann constant k B . Also known
as Boltzmann postulate, Eq. (1.188), was rewritten by Einstein as:
Pr(state) ∝ e
S/K B
(1.189)
meaning that a fluctuation away from the equilibrium state, characterized by a (negative) variation of entropy S, is exponentially unlikely. This begins the industry of
fluctuation theories, and imparts momentum to stochastic modeling in Physics.
The distribution f appearng in the Boltzmann equation is a probability distribution
obtained by projecting down the probability distribution for N particles on the 2d N
exceedingly high dimensional phase space, to the 2d dimensional 1-particle space.
17
In other words,
f (q, p; t)dqdp
(1.190)
represents the probability of finding at time t one particle in the elementary volume
dqdp, i.e. to find one particle within a volume of size dq around q, with momentum
within a set of size dp around p. Furthermore, probability is intended here as the
17 The 2d-dimensional space of a single particle positions and velocities is called μ-space, to distinguish it from the 6N dimensional -space, which is the phase space.