2 On the Foundational Principles of Statistical Mechanics
95
2.7 Arrow of Time: Irreversibility
The macroscopic irreversibility has been visualized as the ‘arrow of time’, which
governs all macroscopic phenomena. Microscopic reversibility and macroscopic irreversibility seem incompatible to each other, which has bothered physicists for ages.
In the history of the development of statistical mechanics, non-equilibrium studies
represented by molecular kinetics came first, followed by equilibrium studies. Many
details about microscopic processes such as collision are involved in molecular kinetics. In his famous recursion theorem Poincaré proved that a finite mechanical system
will return to a state arbitrarily close to its initial state [14]. This was like a bug
that disturbed Boltzmann throughout his whole life. The axiomatic representation of
equilibrium statistical distribution by Gibbs earned a great accolade (his monograph
was highly praised by Poincaré just after publication.) In Gibbs’ theory the bug is
unseen, but not removed. In fact, the microscopic reversibility is associated with the
evolution of molecular orbits while the macroscopic irreversibility is associated with
the evolution of distributions; the two need not contradict each other at all.
In the summer of 1953, Fermi, Pasta and Ulam (FPU) conducted a numerical
simulation of a one-dimensional lattice of nonlinearly coupled oscillators on the
MANIAC, an eirly computer just available then [15]. It was expected that energy
would equalized among different modes, but only the phenomenon of Poincaré recursion was seen. This is called TPU paradox. (The FPU problem should be called the
FPUT problem to acknowledge the contribution of Tsingou who programmed the
MANIAC simulation.) The experiment designed by Fermi was to see molecular
orbits, so of course only the Poincaré recursion was seen. If one wants to see the
evolution of distribution, a set of initial states should be taken, and mixing among
orbits allowed.
For exploring the connection between molecular orbits and distributions one
approach starts from statistical mechanics, another from nonlinear dynamics. As for
how to deal with nonlinear dynamics in the language of distribution, Kolmogorov
claimed that only the invariant distribution which survives after the strength of the
noise exerting on a system approaches zero is meaningful in physics. As a simple
example of reversible dynamics, the baker map is defined as
(x
, y
) =
(2x,
1
2
y),
for 0 ≤ x <
1
2
,
(2x − 1,
1
2
(y + 1)), for
1
2
≤ x < 1.
It is worth demonstrating Kolmogorov’s idea with the baker map.
2.8 Time Evolution of the Distribution Function
As mentioned above, a state of statistical mechanics means a distribution function
in the microscopic phase space. Statistical mechanics ultimately has to deal with the
95
2.7 Arrow of Time: Irreversibility
The macroscopic irreversibility has been visualized as the ‘arrow of time’, which
governs all macroscopic phenomena. Microscopic reversibility and macroscopic irreversibility seem incompatible to each other, which has bothered physicists for ages.
In the history of the development of statistical mechanics, non-equilibrium studies
represented by molecular kinetics came first, followed by equilibrium studies. Many
details about microscopic processes such as collision are involved in molecular kinetics. In his famous recursion theorem Poincaré proved that a finite mechanical system
will return to a state arbitrarily close to its initial state [14]. This was like a bug
that disturbed Boltzmann throughout his whole life. The axiomatic representation of
equilibrium statistical distribution by Gibbs earned a great accolade (his monograph
was highly praised by Poincaré just after publication.) In Gibbs’ theory the bug is
unseen, but not removed. In fact, the microscopic reversibility is associated with the
evolution of molecular orbits while the macroscopic irreversibility is associated with
the evolution of distributions; the two need not contradict each other at all.
In the summer of 1953, Fermi, Pasta and Ulam (FPU) conducted a numerical
simulation of a one-dimensional lattice of nonlinearly coupled oscillators on the
MANIAC, an eirly computer just available then [15]. It was expected that energy
would equalized among different modes, but only the phenomenon of Poincaré recursion was seen. This is called TPU paradox. (The FPU problem should be called the
FPUT problem to acknowledge the contribution of Tsingou who programmed the
MANIAC simulation.) The experiment designed by Fermi was to see molecular
orbits, so of course only the Poincaré recursion was seen. If one wants to see the
evolution of distribution, a set of initial states should be taken, and mixing among
orbits allowed.
For exploring the connection between molecular orbits and distributions one
approach starts from statistical mechanics, another from nonlinear dynamics. As for
how to deal with nonlinear dynamics in the language of distribution, Kolmogorov
claimed that only the invariant distribution which survives after the strength of the
noise exerting on a system approaches zero is meaningful in physics. As a simple
example of reversible dynamics, the baker map is defined as
(x
, y
) =
(2x,
1
2
y),
for 0 ≤ x <
1
2
,
(2x − 1,
1
2
(y + 1)), for
1
2
≤ x < 1.
It is worth demonstrating Kolmogorov’s idea with the baker map.
2.8 Time Evolution of the Distribution Function
As mentioned above, a state of statistical mechanics means a distribution function
in the microscopic phase space. Statistical mechanics ultimately has to deal with the