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Page 1arXiv:math-ph/0008018 v1 7 Aug 2000
Change, Time and Information Geometry
∗
Ariel Caticha
Department of Physics, University at Albany-SUNY,
Albany, NY 12222, USA.
†
SKIPPING ALL PREREQUISETS OF THE ABSTRACT AND INTRODUCTION LEADING
(8)
This is the measure of distinguishability we seek; a small value of dℓ
2
means
the points A and A + dA are difficult to distinguish. The g
αβ
are recognized as
elements of the Fisher information matrix [3
].
Up to now no notion of distance has been introduced on the space of states.
Normally one says that the reason it is difficult to distinguish between two
5
Page 6points in say, the real space we seem to inhabit, is that they happen to be
too close together. It is very tempting to invert the logic and assert that the
two points A and A + dA must be very close together whenever they happen
to be difficult to distinguish. Thus it is natural to interpret g
αβ
as a metric
tensor [
4]. It is known as the Fisher-Rao metric, or the information metric. A
disadvantage of these heuristic arguments is that they do not make explicit a
crucial property of the Fisher-Rao metric, except for an overall multiplicative
constant this Riemannian metric is unique [5
][6].
To summarize: the very act of assigning a probability distribution p(xA) to
each point A in the space of states, automatically provides the space of states
with a metric structure.
The coordinates A are quite arbitrary, they need not be the expected values
〈a
α
〉. One can freely switch from one set to another. It is then easy to check
that g
αβ
are the components of a tensor, that the distance dℓ
2
is an invariant,
a scalar. Incidentally, dℓ
2
is also dimensionless. There is, however, one special
coordinate system in which the metric takes a form that is particularly simple.
These coordinates are the expected values themselves, A
α
= 〈a
α
〉. In these
coordinates,
g
αβ
= −
∂
2
S(A)
∂A
α
∂A
β
(9)
with S(A) given in Eq.(5
) and the covariance is not manifest.
3 Intrinsic dynamics and time
Our basic dynamical principle is that small changes from one state to another
are possible and do, in fact, happen. We do not explain why they happen but, if
we are given the valuable piece of information that some change will occur, we
can then venture a guess, make a prediction as to what the most likely change
will be.
Before giving mathematical expression to this principle we note that large
changes are assumed to be the cumulative result of many small changes. As
the system moves it follows a continuous trajectory in the space of states. We
almost hesitate to call this self-evident fact an assumption, but as the example
of quantum theory shows, trajectories need not exist.
Thus in order to go from one state to another the system will have to move
through intermediate states; in order to change by a distance 2dℓ the system
must have first changed by a distance dℓ.
Suppose the system was in the state A
α
old
= A
α
and that it changes by a small
amount dℓ to a nearby state. We have to select one new state A
α
new
= A
α
+dA
α
from among those that lie on the surface of an n
A
-dimensional sphere of radius
dℓ centered at A
α
. This is precisely what the ME principle was designed to do
[
7], namely, to select a preferred probability distribution from within a specified
given set. The only difference with more conventional applications of the ME
principle is the geometrical nature of the constraint.
6
Page 7We want to maximize S(A
α
+ dA
α
) under variations of dA
α
constrained by
g
αβ
dA
α
dA
β
= dℓ
2
. The notation dA
α
=
˙
A
α
dℓ is slightly more convenient; we
maximize S(A
α
+
˙
A
α
dℓ) under variations of
˙
A
α
constrained by
g
αβ
˙
A
α
˙
A
β
= 1 .
(10)
Introducing a Lagrange multiplier ω,
δ [S(A
α
+
˙
A
α
dℓ) − ω (g
αβ
˙
A
α
˙
A
β
− 1)] = 0,
(11)
we get
[ ∂S
∂A
α
dℓ − 2ω g
αβ
˙
A
β
] δ ˙A
α
= 0 .
(12)
Therefore, writing ω = σ dℓ/2, we get
˙
A
α
=
1
σ
g
αβ
∂S
∂A
β
,
(13)
where g
αβ
is the inverse of g
αβ
. This is our main result; it can be rewritten as
˙
A
α
=
1
σ
λ
α
(14)
where the vector λ
α
,
λ
α
= g
αβ
∂S
∂A
β
,
(15)
is the entropy gradient. The interpretation is clear, the system moves along the
entropy gradient.
This seems such an obvious result that it can hardly be new. Notice, however,
the gradient vector refers to the direction in which there is a maximum increase
per unit distance; one cannot talk about the gradient vector without having
first introduced a metric. The differential form defined by the derivatives S
,β
=
∂S/∂A
β
= λ
β
, the gradient one-form, does not define a direction; it is not by
itself sufficient to define the trajectory.
The physical significance of the Lagrange multiplier σ derives from the con-
straint Eq.(
10) which, using Eq.(14), can be written as
λ
α
λ
α
= σ
2
or σ = (λ
α
λ
α
)
1/2
,
(16)
σ is the magnitude of the entropy gradient. Furthermore, from this and Eq.(
14),
we get dS = λ
α
˙
A
α
dℓ = σdℓ, or
σ =
dS
dℓ
.
(17)
σ is the rate of entropy increase along the trajectory.
The main result, Eq.(1
4), determines the trajectory followed by the system.
It determines the tangent vector
˙
A
α
= dA
α
/dℓ, but not the “velocity” dA
α
/dt.
To fix this something must be said about the universe external to the system,
something that relates the distance ℓ relative to the external time t. This is,
7
Page 8in part, the role normally played by the Hamiltonian, it fixes the evolution of
a system relative to external clocks. If we cannot appeal to such information
(presumably because we do not have it, but perhaps because we just do not want
to), then the only “time” available must be internal to the system, intrinsic to
the geometry of the space of states.
One convenient choice of intrinsic time τ is the distance ℓ itself, or dτ = dℓ.
Intrinsic time is change. The equation of motion is very simple: the trajectory,
A
α
= A
α
(τ), is along the entropy gradient, and the system moves with unit
velocity,
˙
A
α
˙
A
α
= 1, or dℓ/dτ = 1.
The absolute speed dℓ/dt remains unknown. Interestingly, there is no guar-
antee that τ will elapse relative to our own external t, we could have a situation
with dτ/dt = 0. A pile of sand could, if left alone, just stay at A
α
(τ
0
) forever;
its intrinsic time τ has stopped at τ
0
. The pile does not change, because it did
not have (intrinsic) time to change. (One can play endless word games here.)
However, should a measurement of one of the variables, for example A
1
,
indicate a change from the value A
1
(τ
0
) to the value that one would normally
associate with another state along the trajectory, say the value A
1
(τ
1
) at the
later time τ
1
, then one is immediately led to infer that the system has moved
along the trajectory. Most probably all the other variables have also changed
from A
α
(τ
0
) to A
α
(τ
1
). In this case the variable A
1
(τ) is playing the role of
an internal clock. The variable A
1
is a good clock provided one can invert
A
1
= A
1
(τ), to get τ = τ(A
1
). Then, the changes in all other variables A
α
=
A
α
[τ(A
1
)] = A
α
(A
1
) can be referred, correlated to the change in A
1
. We see
that the loss of predictive power due to the unknown absolute speed dℓ/dt is
quite minimal, particularly for high dimensionality (large n
A
).
At this point one could agree that the notion of τ is useful, perhaps even
elegant. But are we justified in calling it time? Perhaps these are mere word
games, but if we do call τ time, then being a distance it provides us with
a model of duration. Furthermore, the very definite ordering of states along
the trajectory A
α
(τ) provides a realization of a temporal order. Finally, the
dynamics is intrinsically asymmetric; the trajectory is intrinsically oriented.
There is one direction in which entropy increases providing a clear distinction
between earlier and later. So this is our answer: we are justified in calling τ
time, because if we do, then we have a neat model, an explanation for temporal
order, for time asymmetry, and for duration. What better reasons do we need?
We close this section with the observation that the system does not follow a
geodesic in the space of states. From Eq.(
14) we can show that the acceleration
vector, given by the absolute derivative (we assume a Riemannian geometry,
with the Levi-Civita connection)
D
˙
A
α
dτ
=
˙
A
α
;β
˙
A
β
= g
αβ
f
βγ
˙
A
γ
,
(18)
does not vanish. The “thermodynamic force” resembles the Lorentz force law in
electrodynamics. The “field strength” tensor f
αβ
, given by f
αβ
=
˙
A
α;β
−
˙
A
β;α
,
is antisymmetric as needed to preserve the unit magnitude of the velocity
˙
A
α
.
8
Page 94 Reciprocal relations
The standard theory of irreversible thermodynamics, due to Onsager [1
3], is
based on the usual postulates of equilibrium thermostatics supplemented by the
additional postulate that the microscopic laws of motion are symmetric under
time reversal. A brief ouline is the following.
As the system moves along its trajectory entropy increases at a rate
dS
dt
=
∂S
∂A
α
dA
α
dt
= λ
α
dA
α
dt
(19)
relative to the external time t; the variables λ
α
are called thermodynamic forces,
and dA
α
/dt are called fluxes. In this theory linear relations between fluxes and
forces are postulated,
dA
α
dt
= L
αβ
λ
β
,
(20)
for which there is abundant experimental evidence, at least close to thermody-
namic equilibrium.
The significance of these relations lies in that they postulate crossed connec-
tions between a flux of type α and a force of type β, and vice versa. (Thus, a
temperature gradient will not just generate an heat current; it may also gener-
ate electric currents, matter flows, and so on.) The strength of these effects is
measured by the phenomenological Onsager coefficients L
αβ
. The central result
of the theory is the reciprocal relation between these crossed effects. The reci-
procity theorem, proved by Onsager on the basis of microscopic reversibility,
states that the matrix of phenomenological coefficients is symmetric
L
αβ
= L
βα
.
(21)
The intrinsic dynamics discussed in the previous section also leads to recip-
rocal relations. The equation of motion, Eq.(1
4), gives
dA
α
dt
=
dτ
dt
dA
α
dτ
=
dτ
dt
1
σ
g
αβ
λ
β
.
(22)
This allows us to identify the Onsager coefficients as
L
αβ
=
dτ
dt
1
σ
g
αβ
.
(23)
These coefficients are not constants, they vary along the trajectory, L
αβ
=
L
αβ
(A).
What is interesting here is that their symmetry follows from the symmetry of
the metric tensor. No hypothesis about microscopic reversibility was needed; in
fact, microscopic dynamics was not mentioned at all. In addition, the validity
of Eq.(
22) is not restricted to the immediate vicinity of equilibrium. To the
extent that the variables A are the right variables to describe phenomena far
from equilibrium, the reciprocal relations should still hold.
9
Page 105 Dynamics constrained by conservation
Beyond the fact that changes happen, perhaps the most common additional
information that one can have about an irreversible process is that some quan-
tities are conserved. As an illustrative example we consider two systems that
are allowed to exchange some conserved quantities and evolve towards equilib-
rium. To fix ideas we could think of an ideal gas filling two vessels at different
temperatures and chemical potentials. Once the two vessels are connected, for
example by a tube, a little hole, or a a porous plug, matter and energy will flow
until equilibrium is reached.
To keep this as simple as possible we assume the experimental conditions
are such that throughout the process the two systems remain homogeneous
and independent. The first system is described by variables A
α
, the second
is described by primed variables A
′α
, and the entropies, given by Eq.(
5), are
additive
S
T
(A, A
′
) = S(A) + S
′
(A
′
).
(24)
Since the quantities A are conserved the dynamics is constrained by A
′
= A
T
−A,
with A
T
fixed, or
˙
A
′
= −
˙
A. The conservation constraint could be incorporated
using Lagrange multipliers; for this simple example it is just as easy to eliminate
A
′
.
In our ideal gas example, the variables could be energy, A
1
= E, and number
of molecules, A
2
= N. This crucial part in setting the problem, choosing the
description, is the one most likely to go wrong. If the hole coupling the two
vessels is too large, the ME predictions below will fail. The failure is not to
be blamed in the ME method, but on the choice of variables: the pair E, N
is not enough to codify the relevant information of say, a turbulent flow. The
same remark applies if the connecting porous plug is such that heat can be
easily exchanged but there is resistance to matter flow. In this case additional
variables are needed, perhaps describing the physical state of the plug and the
gas in it.
Suppose the system was in the state A and that it changes by a small amount
dτ to a nearby state. To select one new state A +
˙
Adτ from among those that
lie on the surface of a sphere of radius dτ centered at A, we maximize
S
T
(A +
˙
Adτ) = S(A +
˙
Adτ) + S
′
(A
T
− A −
˙
Adτ)
(25)
under variations of
˙
A
α
constrained by
g
αβ
˙
A
α
˙
A
β
= 1 .
(26)
where the Fisher-Rao metric, Eq.(9
), is given by
g
αβ
= −
∂
2
∂A
α
∂A
β
(S(A) + S
′
(A
T
− A)) .
(27)
The result is
dA
α
dτ
=
1
σ
g
αβ
(λ
β
− λ
′
β
) ,
(28)
10
Page 11where σ is the rate of entropy production σ = dS
T
/dτ. The system evolves
until the conjugate variables λ are equalized.
6 Final remarks
The main conclusion is simple: unless there is positive evidence to the contrary,
our best prediction is that the system evolves along the entropy gradient. What
is perhaps not so trivial is that, unlike other conventional forms of dynamics,
this intrinsic dynamics does not require an additional postulate. It is the unique
dynamics that follows from the maximum entropy principle and nothing else.
Another nontrivial aspect is that the model supplies its own notion of time.
Since the irreversible macroscopic motion is not explained in terms of a reversible
microscopic motion there is no need to explain irreversibility, this question never
arises. Similarly, there is no need to explain the second law of thermodynamics;
it is the second law (in the form of the ME axioms) that explains everything
else.
These ideas can be explored further in a number of directions. There is,
for example, the relation with other theories of irreversible processes, such as
the equations of hydrodynamics. Another possibility is to extend the theory to
account for fluctuations and diffusion. The intrinsic dynamics proposed above
is deterministic, but to the extent that the ME principle does not completely
rule out distributions of lower entropy
[7], fluctuations about equilibrium and
about the deterministic motion are possible.
Perhaps the most intriguing question to pursue stems from the possibility
of deriving dynamics from purely entropic arguments. This is clearly valuable
in areas where the microscopic dynamics may be too far removed from the phe-
nomena of interest, say in biology or ecology, or where it may just be unknown
or perhaps even inexistent, as in economics. One could argue that these theo-
ries would be phenomenological as opposed to fundamental, that within physics
the search for a fundamental mechanics would still be left open. However, in
previous work we have shown [2
3] that entropic arguments do account for a
substantial part of the formalism of quantum mechanics, a theory that is pre-
sumably fundamental. Perhaps the fundamental theories of physics are not so
fundamental; they are just consistent, objective ways to manipulate information.
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13
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