40
L. Rondoni
1.6 Particle Systems in Phase Space
Let us now relate the microscopic atomic description of matter to the macroscopic
thermodynamc description. The first question, in view of the observations made in the
Introduction, is which microscopic model should one use for such a task. A priori
it is not possible to tell, therefore we begin with classic Hamiltonian mechanics.
Why? Because it has proven extremely successful in describing a huge variety of
phenomena that occur at the scales of our daily life, and well beyond, reaching
astrophysical objects. This, at least, was the opinion expressed by Laplace, when
stating [17]:
We may regard the present state of the universe as the effect of its past and the cause of
its future. An intellect which at a certain moment would know all forces that set nature in
motion, and all positions of all items of which nature is composed, if this intellect were
also vast enough to submit these data to analysis, it would embrace in a single formula the
movements of the greatest bodies of the universe and those of the tiniest atom; for such an
intellect nothing would be uncertain and the future just like the past would be present before
its eyes. 10
Now, rather than describing a moderate number of macroscopic objects, for which
classical mechanics has proven so successfull, we want to describe an exceedingly
large assembly of very small objects. This may be a cause of concern; as usual, we
will judge the validity of this approach from the results it will produce.
Given a system of N particles each having d degrees of freedom, we introduce the
space of “microscopic phases”, or simply phase space, M ⊂ IR
2d N , and its points
= (q, p) ∈ M, where q represents the d N -dimensional configuration vector, and
p the d N -dimensional vector of momenta. It is assumend that each fully represents the microscopic state of the particle system. Then, in the case of Hamiltonian
dynamics, there exists an energy function H = H ((), such that the time evolution
is given by:
˙
q =
∂ H
∂p
, ˙
p = −
∂ H
∂q
(1.163)
We denote by S
t
: M → M the evolution operator for a time t ∈ R, meaning that
S
t
∈ M is the microstate at time t, if it was at time 0.
To connect with the macroscopic level of observation, one introduces the notion of
phase function, O : M → IR, and, considering that a measurement takes a positive
time, one assumes that it yields a time average of one phase function, also called
observable. If the measurement takes a time τ , and it starts when the microstate is
, the result of the measurement is:
O 0,τ (() =
1
τ
τ
0
O(S
t
) dt
(1.164)
10 Note, however, that considering that such an intellect is out of our reach, the best approach to
nature is probabilistic.
L. Rondoni
1.6 Particle Systems in Phase Space
Let us now relate the microscopic atomic description of matter to the macroscopic
thermodynamc description. The first question, in view of the observations made in the
Introduction, is which microscopic model should one use for such a task. A priori
it is not possible to tell, therefore we begin with classic Hamiltonian mechanics.
Why? Because it has proven extremely successful in describing a huge variety of
phenomena that occur at the scales of our daily life, and well beyond, reaching
astrophysical objects. This, at least, was the opinion expressed by Laplace, when
stating [17]:
We may regard the present state of the universe as the effect of its past and the cause of
its future. An intellect which at a certain moment would know all forces that set nature in
motion, and all positions of all items of which nature is composed, if this intellect were
also vast enough to submit these data to analysis, it would embrace in a single formula the
movements of the greatest bodies of the universe and those of the tiniest atom; for such an
intellect nothing would be uncertain and the future just like the past would be present before
its eyes. 10
Now, rather than describing a moderate number of macroscopic objects, for which
classical mechanics has proven so successfull, we want to describe an exceedingly
large assembly of very small objects. This may be a cause of concern; as usual, we
will judge the validity of this approach from the results it will produce.
Given a system of N particles each having d degrees of freedom, we introduce the
space of “microscopic phases”, or simply phase space, M ⊂ IR
2d N , and its points
= (q, p) ∈ M, where q represents the d N -dimensional configuration vector, and
p the d N -dimensional vector of momenta. It is assumend that each fully represents the microscopic state of the particle system. Then, in the case of Hamiltonian
dynamics, there exists an energy function H = H ((), such that the time evolution
is given by:
˙
q =
∂ H
∂p
, ˙
p = −
∂ H
∂q
(1.163)
We denote by S
t
: M → M the evolution operator for a time t ∈ R, meaning that
S
t
∈ M is the microstate at time t, if it was at time 0.
To connect with the macroscopic level of observation, one introduces the notion of
phase function, O : M → IR, and, considering that a measurement takes a positive
time, one assumes that it yields a time average of one phase function, also called
observable. If the measurement takes a time τ , and it starts when the microstate is
, the result of the measurement is:
O 0,τ (() =
1
τ
τ
0
O(S
t
) dt
(1.164)
10 Note, however, that considering that such an intellect is out of our reach, the best approach to
nature is probabilistic.