44
L. Rondoni
This requires a rule governing time dependent probabilities in phase space. What
rule? If probability has to adhere to the behavior of material objects, its evolution
should presumably be related to that of matter, and matter moves according to the
laws of mechanics. However, unlike material objects, a chunk of probability does not
have inertia, and there are no forces that push it like Newton’s second law prescribes;
even the space in which probability may somehow move is quite peculiar compared
to the three-dimensional space where material objects move. The only property of
mass that probability seems compelled to share is that it should be conserved.
14 One
may then argue that probability moves in phase space like a fluid moves in real
space; in other words, one may assume that sets of phase space points –that move
in phase space according to dynamical laws such as Eq. (1.163)– carry probability
with themselves, like fluid elements carry mass with themselves. To formalize this
idea, let us concisely write the (not necessarily Hamiltonian) evolution equation for
phases as:
˙
= G((), , ∈ M
(1.168)
where G : M → M is a vector field, and let us endow M with a probability density
f . The above assumption amounts to postulating that f obeys a continuty equation
in M:
∂ f
∂t
= −∇ · ( f G) = f
f
; or
d f
dt
= − f
(1.169)
where
(() = ∇ · G|
(1.170)
is the phase space volume variation rate at , and
f
(() = −G(() · ∇ ln f | − (()
(1.171)
is known as dissipation function. Equation (1.169) is the generalized Liouville equation, which reduces to
d f
dt
= 0
(1.172)
in the case of Hamiltonian dynamics, because:
(() =
i
∂
∂q i
∂ H
∂ p i
−
∂
∂ p i
∂ H
∂q i
= 0
Denoting the time integrals and averages of a phase function O by:
14 For instance, what would it mean that the probability of the set containing all possible events
turns larger than 1? Or that it gets smaller than 1? It would indicate that at some point one was not
or is not really treating all possible events.
L. Rondoni
This requires a rule governing time dependent probabilities in phase space. What
rule? If probability has to adhere to the behavior of material objects, its evolution
should presumably be related to that of matter, and matter moves according to the
laws of mechanics. However, unlike material objects, a chunk of probability does not
have inertia, and there are no forces that push it like Newton’s second law prescribes;
even the space in which probability may somehow move is quite peculiar compared
to the three-dimensional space where material objects move. The only property of
mass that probability seems compelled to share is that it should be conserved.
14 One
may then argue that probability moves in phase space like a fluid moves in real
space; in other words, one may assume that sets of phase space points –that move
in phase space according to dynamical laws such as Eq. (1.163)– carry probability
with themselves, like fluid elements carry mass with themselves. To formalize this
idea, let us concisely write the (not necessarily Hamiltonian) evolution equation for
phases as:
˙
= G((), , ∈ M
(1.168)
where G : M → M is a vector field, and let us endow M with a probability density
f . The above assumption amounts to postulating that f obeys a continuty equation
in M:
∂ f
∂t
= −∇ · ( f G) = f
f
; or
d f
dt
= − f
(1.169)
where
(() = ∇ · G|
(1.170)
is the phase space volume variation rate at , and
f
(() = −G(() · ∇ ln f | − (()
(1.171)
is known as dissipation function. Equation (1.169) is the generalized Liouville equation, which reduces to
d f
dt
= 0
(1.172)
in the case of Hamiltonian dynamics, because:
(() =
i
∂
∂q i
∂ H
∂ p i
−
∂
∂ p i
∂ H
∂q i
= 0
Denoting the time integrals and averages of a phase function O by:
14 For instance, what would it mean that the probability of the set containing all possible events
turns larger than 1? Or that it gets smaller than 1? It would indicate that at some point one was not
or is not really treating all possible events.