1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
43
ranges. If this condition had to be verified by all possible phase functions, the time
required for a systems of many degrees of freedom could be so hugely long, to make
the ergodic hypothesis physically irrelevant. Luckily, macroscopic observations only
concern a handful of variables, which in addition are quite well behaved. This explains
why ergodicity is continually and succesfully adopted.
In some cases, this reasoning can be rigorously justified. The approach developed
by Khinchin for rarefied gases does that, explaining that phase space subleties are
irrelevant, compared to the fact that [21]:
(a) macroscopic systems are made of very many particles: N ≫ 1;
(b) only several and special phase functions are physically relevant;
(c) it does not matter if ensemble averages disagree with time averages on a limited
sets of trajectories.
For rarefied gases, the relevant phase functions are sums of molecular contributions, f (() =
N
n=1 f n (q n , p n ), where (q n , p n ) is the vector of configurations and
momentum of the n-th particle. These functions are appropriate for the pressure,
temperature and density of rarefied gases, whose energy can also be expressed as
H =
N
n=1 H n (q n , p n ), because interactions among particles are energetically negligble.
13 Then, in the microcanonical case, which in this case means a uniform probability distribution in M, Khinchin proved the validity of the following relation:
Prob
| f − − f |
|| f |
≥ K 1 N
−1/4
≤ K 2 N
−1/4
,
(1.167)
where K 1 and K 2 are positive constants, f is the time average of f , and f its phase
space average in the microcanonical ensemble. In other words, the probability (in the
microcanonical sense) that time averages differ by a small amount from the phase
space averages is small if N is large: the larger N the smaller the probability of even
smaller differences.
In this framework, the physically relevant ergodicity follows by and large from
the N ≫ 1 condition, combined with the validity of the law of large numbers, which
make the sum variables practically constant, and the exploration of their range fast.
The details of the microscopic dynamics, including transtivity or lack of transitivity
result irrelevant, while ensembles, i.e. probabilities in phase space turn considerably
useful; even though probability, per se is an immaterial and abstract mathematical
notion, it becomes “real”, in some sense, under the above conditions.
To understand more deeply when and why probability can be treated as real,
which arguably makes it the most useful mathematical tool in Physics, let us consider evolving states, rather than stationary states, thus challenging Remark 1.6.1.
If a stationary probability distribution correctly represents an equilibrium state, an
evolving probability might, perhaps under different conditions, represent an evolving
state.
13 Interactions are however essential for the condition of LTE, hence for the existence of thermodyamic properties, to be established.
43
ranges. If this condition had to be verified by all possible phase functions, the time
required for a systems of many degrees of freedom could be so hugely long, to make
the ergodic hypothesis physically irrelevant. Luckily, macroscopic observations only
concern a handful of variables, which in addition are quite well behaved. This explains
why ergodicity is continually and succesfully adopted.
In some cases, this reasoning can be rigorously justified. The approach developed
by Khinchin for rarefied gases does that, explaining that phase space subleties are
irrelevant, compared to the fact that [21]:
(a) macroscopic systems are made of very many particles: N ≫ 1;
(b) only several and special phase functions are physically relevant;
(c) it does not matter if ensemble averages disagree with time averages on a limited
sets of trajectories.
For rarefied gases, the relevant phase functions are sums of molecular contributions, f (() =
N
n=1 f n (q n , p n ), where (q n , p n ) is the vector of configurations and
momentum of the n-th particle. These functions are appropriate for the pressure,
temperature and density of rarefied gases, whose energy can also be expressed as
H =
N
n=1 H n (q n , p n ), because interactions among particles are energetically negligble.
13 Then, in the microcanonical case, which in this case means a uniform probability distribution in M, Khinchin proved the validity of the following relation:
Prob
| f − − f |
|| f |
≥ K 1 N
−1/4
≤ K 2 N
−1/4
,
(1.167)
where K 1 and K 2 are positive constants, f is the time average of f , and f its phase
space average in the microcanonical ensemble. In other words, the probability (in the
microcanonical sense) that time averages differ by a small amount from the phase
space averages is small if N is large: the larger N the smaller the probability of even
smaller differences.
In this framework, the physically relevant ergodicity follows by and large from
the N ≫ 1 condition, combined with the validity of the law of large numbers, which
make the sum variables practically constant, and the exploration of their range fast.
The details of the microscopic dynamics, including transtivity or lack of transitivity
result irrelevant, while ensembles, i.e. probabilities in phase space turn considerably
useful; even though probability, per se is an immaterial and abstract mathematical
notion, it becomes “real”, in some sense, under the above conditions.
To understand more deeply when and why probability can be treated as real,
which arguably makes it the most useful mathematical tool in Physics, let us consider evolving states, rather than stationary states, thus challenging Remark 1.6.1.
If a stationary probability distribution correctly represents an equilibrium state, an
evolving probability might, perhaps under different conditions, represent an evolving
state.
13 Interactions are however essential for the condition of LTE, hence for the existence of thermodyamic properties, to be established.