1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
77
One may still conclude that such a detailed description is not necessary, and not
even useful; in fact, as discussed earlier, the average picture is much more sensible
than the detailed picture. In this case, we may say that we integrate the equations of
motion first, and we average after:
t
0
˙
v(t
; x 10 , v 10 )dt
−→ E
⎡
⎣
t
0
˙
v(t
; x 10 , v 10 )dt
⎤
⎦ =
dμ(x 10 , v 10 )
t
0
dt
˙
v(t
; x 10 , v 10 )
(1.275)
where μ is the probability distribution of the initial conditions of the test particle.
There is a practically almost equivalent point of view. One admits at start that
only statistical properties make sense, and does not bother considering deterministic
equations. After all, neither the deterministic nor the stochastic nature of a given
phenomenon can be empirically validated; it is only a matter of deciding which of
the available models better describes that phenomenon. Therefore, one may adopt
the stochastic description not because of (possibly partial) ignorance of the initial
state, but as a fully fundamental approach. In this case, one derives equations for
probabilities and expectation values, and eveolves them. Then it is like inverting the
order of the integrals in Eq. (1.275):
E t [v] =
t
0
ds
d
ds
dμ s (v) v =
t
0
ds ˙
E s [v]
(1.276)
where at time s the random variable v is distributed as prescribed by μ s , and no
claim is made on the existence of ˙
v. Within the deterministic view dealing with a
finite number of particles N whose accelarations exist, averaging with respect to μ s
amounts to sum over the N values of v and divide by N , therefore the two approaches
are interchangeable.
Appendix 2: Further Readings
It is impossible to do justice to so many different fields in a single paper. Therefore,
we stop here, but we report a selection of readings, to complement the bibliography
related to the content of the paper. Even this list is largely incomplete, but it provides further food for thought. I apologize for all unforgivable omissions. The list is
presented in alphabetical order.
1. Paolo Adamo, Roman Belousov, and Lamberto Rondoni, Fluctuation-Dissipation
and Fluctuation Relations: From Equilibrium to Nonequilibrium and Back, in
A. Vulpiani et al. (eds.), Large Deviations in Physics, Lecture Notes in Physics
885, Springer-Verlag Berlin Heidelberg 2014
2. Ping Ao, Potential in stochastic differential equations: novel construction, Journal of physics A: mathematical and general 37 (3), L25
77
One may still conclude that such a detailed description is not necessary, and not
even useful; in fact, as discussed earlier, the average picture is much more sensible
than the detailed picture. In this case, we may say that we integrate the equations of
motion first, and we average after:
t
0
˙
v(t
; x 10 , v 10 )dt
−→ E
⎡
⎣
t
0
˙
v(t
; x 10 , v 10 )dt
⎤
⎦ =
dμ(x 10 , v 10 )
t
0
dt
˙
v(t
; x 10 , v 10 )
(1.275)
where μ is the probability distribution of the initial conditions of the test particle.
There is a practically almost equivalent point of view. One admits at start that
only statistical properties make sense, and does not bother considering deterministic
equations. After all, neither the deterministic nor the stochastic nature of a given
phenomenon can be empirically validated; it is only a matter of deciding which of
the available models better describes that phenomenon. Therefore, one may adopt
the stochastic description not because of (possibly partial) ignorance of the initial
state, but as a fully fundamental approach. In this case, one derives equations for
probabilities and expectation values, and eveolves them. Then it is like inverting the
order of the integrals in Eq. (1.275):
E t [v] =
t
0
ds
d
ds
dμ s (v) v =
t
0
ds ˙
E s [v]
(1.276)
where at time s the random variable v is distributed as prescribed by μ s , and no
claim is made on the existence of ˙
v. Within the deterministic view dealing with a
finite number of particles N whose accelarations exist, averaging with respect to μ s
amounts to sum over the N values of v and divide by N , therefore the two approaches
are interchangeable.
Appendix 2: Further Readings
It is impossible to do justice to so many different fields in a single paper. Therefore,
we stop here, but we report a selection of readings, to complement the bibliography
related to the content of the paper. Even this list is largely incomplete, but it provides further food for thought. I apologize for all unforgivable omissions. The list is
presented in alphabetical order.
1. Paolo Adamo, Roman Belousov, and Lamberto Rondoni, Fluctuation-Dissipation
and Fluctuation Relations: From Equilibrium to Nonequilibrium and Back, in
A. Vulpiani et al. (eds.), Large Deviations in Physics, Lecture Notes in Physics
885, Springer-Verlag Berlin Heidelberg 2014
2. Ping Ao, Potential in stochastic differential equations: novel construction, Journal of physics A: mathematical and general 37 (3), L25