情報熱力学と微分幾何学

:VIEMolRs oF SAGAMI

INgTITvTE oF TEcilNoLo{;y

Vol. 12.No.1,1978

Information

Thermodynamics

and

Differential

Geometry

Roman

S.

INGARDEN"

In

1975 inmy leeturein

Ohmihachiman,

then _published

in

Tensor

[1],

I

_proposed

differential-geometrieal

investigation

of the parameter space 'ofquantum

(or

elassieal)

information

thermo-dynamics

with

higher

order temperatures.

Today,

during

my second stay

in

Japan,

I

am able

to _present

here

a _partialsolution of this _problem, anyhow a principalsolution whieh ean

be

performed to the end

for

simplest eases,

but

for

more complieated ones, as usually, requires

yet a

lot

of work and can

be

done

mostly only

in

a qualitative sense.

The

solution

has

been

found

due

to a remarkable

investigation

executed quite reeently in

Sapporo

during

my stay

there

by

Y.

Sato,

K.

Sugawa

and M.

Kawaguchi

[2],

ef.

[3].

In

my _present

leeture

I

will

tryto

be

as elementary as _possibleand will show the merit on the simplest

_physical

examples

not assuming any speeial

knowledge

of _physies. For further

details

I have to refer to the

publications which are in _preparation

[2]

and

[3].

Let

us

first

consider the simplest ease of classical statisticai thermodynamics, that of _an

ideal _gas. We shall eonsider thisease eompletely

by

the method of

information

thermodynamies

whieh seems to

be

not only the simplest one,

but

also mathematically most clear and

_preeise

(for

deeper _physieal foundation of thismethod cf.

[4]).

We ean imagine n _point non-interaeting

parti¢

les

contained

in

a

3-dimensional

domain

A

of volume

V

(A

can

be,

e.g., a cube,

but

not

necessarily). The space is Euelidean and we

do

not consider any outer _physical fieldin this spaee,

just

for

the sake of simplicity.

The

_particlesare of mass m and their _positions and velocities ean

be

_presented

by

means of orthogonal

Cartesian

coordinates in R3,

(1)

q=(q.,)=(q.,(t))cQ,

4=(4.i)=-(ddqi`)=(a..(t))EP,

where

(2)

a==1,2,･･･,n,

i=1,2,3,

Q=]Ax･･･xA=AncR3n,

P=Ran,tER.

Introducing rnomenta

(3)

P==(Pai(t))=(Mdai(t))EP,

we ean write the total energy, as the sum of the

kinetic

and _potentialenergy,

(4)

H'==H(q,p)=K(p)+W(q)=K(p)=2P.2

=.l,

t".,

ZZ'i'

since we can assume that

17V

vanishes

in

Q

(outside

of

Q

it

is

oo).

Now

we may consider a

probability problem in the _phase space 9==QxPcR6",

(9,

ut,_p),pt(n)=1. Assuming that _pthas

the

Radon-Nikodym

derivative

with respect to the

Lebesgue

measure in _q,

p,

we ean write

* Professor,

Institute

of Physics,Nieholas,

Copernieus

University,

ul, Grudziadzka5, Torufi87-100,

Poland.

nt-7 >. Fpaop u he _iv := fi-Jk \opignt

("Mtzmp\k'ra'ilkft)

'e 1977 _¢ 8A rbi5 10 A t･ckH, 9 Jl16m

Nec]

* k\va

pt

ee

12

#

as

le

(5)

"(D) ==

I.f(q,

p)dS"q

dS"p

)O,

DEur,

where

f(p,

_q) isa ut-measurable function on 9 such that

(6)

f(q,p))O

in

9

and

I.f(q,

p)de"q

d3"p=1=pt(9).

We

assume now that

f

maximizes the information

(entropy)

with respect to

Lebesgue

.

measure ln p,a

(7)

_S=

-leI.f

_{lnfdS"a dS"p}

(k

is

the

Boltzrnann

constant) under the eondition that the mean energy

(s)

u=!.Hfds"q

ds"p=

2=,

,i.,

:h't<+oo

is

finite

and

fixed.

That

is

the

assumption of

the

maximum chaos

<indeterminacy)

in

the

phase space

9

under the condition

(8)

whieh expresses the _properties of the environment

being

a

thermostat

with

fixed

mean energy

U.

Using

the

well-known variational proeedure

we caleulate the

first

variational

derivative

of the

functional

(9)

L==:S--}U-

_ic

_{ln Z+1==}

-k!.f(ln

_f+

ln Z+ 2mPk2

T-1)dS"a dS"p

,

where the two

Lagrange

multipliers eorresponding to

(8)

and

(6)

are

denoted

<to

comply with

the usual _physical notation)

by

--T-t and -klnZ+1, respeetively.

We

obtain

(io)

6ivL,=: -le(inf+in Z+

2mPi

T)=O

or

al)

_f==

z-texp

(-2mPi

T)

:=

z-i

exp(-H

£

qT'P))

as the neeessary eondition

for

the extremum. For ealculating the undeterminate constants

T

and

Z

we

have

to use conditions

(8)

and

(6).

From

(6)

we easily obtain using the well-known

formula for the integral of the

Gauss

funetion _{and the condition that}

A

has

volume

V

(12) Z=Z(T)=V"(2nmkT)S"f2.

Therefore,

(13)

f(q,p;

V;

T)=V-"(2TmicT)-S"/2

exp(-2mPk2

T)･

To

calculate

T

we remark that

by

(8)

and

(11)

(14)

U=:-o(:IT)

lnZ(T)==leT2oaT

ln

Z(T).

infbrmation Thermodynamics and DiLtilerentiatGeometry

Using

(12)

we obtain from

(14)

(ls)

U=llttkT or T==;nUk,

in accordance with the so-ealled _principle of equipartition of energy saying that for eaeh

degree

of

freedom

of a _physica! system we _get in mean

licT

_of energy

(our

system has

3n

degrees

of

freedom).

Looking at

f(q,p;

V,

T)

in

(13)

we see that it

depends

on two sets of variables:"micro-scopical" _phase space variables

(q,

p)(9ERe" and "maeroscopical" _{thermodynamical}

parameters

(V,

T)ER2

sueh that

(16)

O<V<+oo,

O<T<+oo,

cf.

(15).

In

his

"catastrophe theory" of _general

(topological)

dynamical

systems

Ren6

Thom

ealls the spaee of the

first

sort of variables "behavior

spaee" and the space of the second

sort of variables "control spaee", whieh corresponds very well to the _physical situation.

In

the multivariate analysis of statistics one speakg about the space of _parameters

(in

our

physical example thisspace

is

2-dimensional),and one

discusses

"distance"

or "divergence"

be-tween

"

statistical hypotheses" represented

by

two _points of sueh _parameter space, cf.

I5],

[6],

[7].

Particularly

interesting

is

the

informational

distance

or

divergence

diseussed

by

Kullback

and

Leibler

[5],

cf. also

[6],

called

by

Renyi

"infermation _gain"

[7]

and

by

Sch16gl

"entropy

production"or "information production"

[8].

Sch16gl

introdueed

this quantity to

thermo-dynamics and

diseussed

its

connection with stability conditions of

Ljapunov

and

Glansdorff-Prigogine

prineipleof irreversiblethermodynamies.

In

our case thisquantity ean

be

written

(17)

_J(1,

2)==

!.(A(q,

p)-h(q, p))ln

ill[gl

;i

dS"q d3"p

,

where

(18)

.tl(q,p)=f(q,p;

Vi,Ti),

.fli(q,p)=f(q,p;

K,Ti)･

Considering

for the moment a _general

(classical)

case we may write in_plaeeof thermodynamical

parameters

V,

T

any number of sueh _parameters, letus call them _u', r=1, ...,m

(so

in

our

¢ase m=2). In 1938 Fisher diseussed his famous information matrix

[91,

ef. also

[5]

and

[6],

which may

be

written

in

thisnotation

(19)

grs=]

!f(la3il.

)(i

_oO.f,

)el3"q

_{ds"p-E(Ool.".f}

Ool.",f),

where E

denotes

the mean value

(expectation

value).

By

means of _g.,we can write,

denoting

u=(iti, ･･-,ieM)

[2],

(20)

J(u,u+elu)==Zg,,durdus=cls2,

r,s

and

C,R.

Rao

[10]

interprets

this as a square of

differential

distanee

defining

locally

the metric of some Riemannian space

(assuming

that _{g=det(g.,):#iO,} as aetually is in the classical case). For more than one _parameter this approaeh was not apparantly developed

reec=\S<\nff

eg

12 ts

zz1Z･

spirit of

differential

_geometry). Diseussing a change of _parameters according to the

group

(12)

u' = _ur(u", -･･,_um ')(r =r

l,

･･･, m)

,

DD(

£31i

ii

ii

2Ue.",))

:,LO,

where u'(ut) are

funetions

of at

least

class

q

(e.g.,

CL.

or

C.),

we easily obtain

Our Oue

(22)

g;･,t=

:[]

,grs s

r,r Oecr'auS

so _g.,

ig

really a tengor

in

the

Riemannian

sense.

Going

over

to

the

contravariant rnetrie

tensor grs,

(23)

ZgregSt==:6rt,

dUr=ZgrsdUS,

s s

we _can write as usually

(24) cts2= _{£ g'Sdu.du,,}

ts

but,

in

general, not

(25)

J(1,

2)

==s2=

Z

_g'e(u.(i)-u.(2))(u, (i)-us(2)) r)s

as was

done

as

definition

of s

by

Mahalanobis

[11]

(divided

by

7n) or

by

Kullback

and

Leibler

[5],

formula

(2.10).

Below

we shall show eases in which,

by

our geometrical interpretation,

(or,

betber

to say,

by

di.t7lerent

dojinition

of s) the

finite

distanee

cannot

be

written

in

form

(25).

Iwould

like

also to point out that, as far as I

know,

_previous

diseussion

of

Riemannian

space

in

statistics and

information

theory was confined either

to

behavior

spaee

(x-space,

when

we

denote

in our case x=(q,p)), or

directly

_postulated, as e.g.

by

Amari

in

"information

space "

[11]

or else

to

one-parameter normal

distributions.

In

the paper

by

Sato,

Sugawa

and

Kawaguehi

[2]

the _probability

density

of the multivariate

normal

distribution

(26)

f(x)=(2iv)-Pi2IXIrii2

exp

(-b(x-p)'

E-i(x-Ft))

,

where p

is

the number of

dimensions

of the x-spaee, _p=(pti,･･･, pt.),

Z==(aij)(i,o'=1,

･'',p),

has

been

considered. In thiscase:

(27)

m=p+"p(p+1).

The

calculations of

the

rnetric and curvature

tensors

have

been

performed

fer

pa=:1

and

_p=2

(m=2

and _m=5), in _the lastcase

by

means of a eomputer. The results are as follows:

1)

p:=1(m:=:2)

(28)

(g.s)==(a62

2aO-,),

Rnn=a-`,

R=-1,

K='-1!2,

g==2a-`#O,

2) _p=2(m=5),

ri=uiian-di2;O,

-86-(29)

(grs)[=

information Thermedynamics and DzlfflerentialGeoenetry

d-ia2,

-d-iai2 O O

O

-A"'ai2 _d'ia"

_O

_O

_O

O

O

SA-2a;2

-zt-2atao22 -!A-'2o?2

O

O

-zt-2ana,, A-2(anan+a;2) -d-Zanoi2

O

O

bd-!ak

-d-2auai2

ld-2ak

=

td-\o

,

R=

-2+l

g2p,

, p==

ol

la,t21g,

,

independent

eoeMeient of the cuvature tensor

(in

the

the

Gauss

totalcurvature, and _p isthe correlation

the metric

is

independent

of _{pt-parameters.}

In

the

in a, while in the seeond ease it

has

a constant

an

iff

_p=O.

It

is

interesting

that

on

the

surfaees

ai2= ±

(ana22)i/2

, A==O , p:=1 ,

domain

is

simply

O<a<+oo,

-oo<tt<+oo.

is essentially

different:

in the firstcase of the

case the space isdevelopable with zero curvature.

insight

we

have

to

discuss

other examples.

be

eonsidered as "inner" or mieroscopie, since : 2e)-2 -(oT)rr' q=(tuT)-2>O, R,,,,=R==K==O . ; (30) g

where

Rim

is

the only

first

ease,

in

the second case the

here),Ris

the

sealar curvature,Kis coeMcient.

We

see

that

in

the

both

cases

first

case

it

isof

theeonstant sealar curvature scalar curvature

in

_pararneters

an, a22,

(31)the

metric

became

singular, geometry has sense

is

(32)

O<a.<+oo

-oe<p,,pt,<+oo.

In

the

first

case this

Going over to our of analogous calculations:

(33)

(g.,)

=;

(3(2

T2)Lin2

...o.

We

see that,

in

spite the

Gibbs

(13)

distributions,

the _geometry constant negative

eurvature, in the second

To

get

deeper

physieal

Fora(classical) linear

frequeney

te only the

first

parameter, m, can tocharacterizes the outer

force

field

(potential

energy) the ideal _gas

(described

in

above correspondence will more

explicitly shown

below)

have now two

thermodynamical parameters

by

ehanging the

length

of a pendulum)

(34)

(grs):=(-(.T)-i

)'

Now we may combine

both

the examples

into

one

by

considering an

ideal

_gas contained

not

in

a rigid-walled vessel

A

with

the

_potentia]

(3s)

w(q)-(O..

itutXxeJit\reet

(36)

vr(q)=

(we

take

here

A

as a cube of side

dependent

3-dimensional

harmonic

eal¢ulatien is

(37)

(g.):=(-s6.n,l:'T')H,

We

now can

_phow

a elassieal

case of the type consider'ed

'in

[1].

for_,Which we assume the

(38)

q=!qraqdp,

q

(39)

Now

we

have

5

Lagrange

in

whieh eross correlation

(39)

will

(40)

(41)

a,2==

(42)

a22:=

<43)

(grt)==

the "ideal gas" of n

and o.

The

end result of

H5snn2StoTT-,)-i),grsn2(evT)L2>O,

Rt2n=R==K=O･

'

ease which

_gives

a

non-zero curvature. This isthe simplest

Let

us consider 'aelassieal system of

1

degree of freedom

following ,conditiong

by

maximization of entropy

(information)

==!pfdqdp,

U,2=!p2fdqdp,

q2=:=!gvaqdp,

U.,

:=

Ipqfdgdp.

coeMcients cr,

P,

_r,ti,e,

but

we ean

introduce

"normal

eoordinates"

disappear.

Finally,

we obtain

denoting

-di=r+6+((r-6)+e)""

,

d,=

r+6-((r-6)2+E)']2 ,

(2d2)-'(6a2+rP2-eaP)-dt-i((afi-ieP)2+(fir-}scr)2)

,

diL'(Oa2+rP-eaP)-(dicl2)-t((a6-bsP)2+(Pr-tEa)2)

, (")The spaee

faetorizes

(45)

(46)

Very

interesting

publication

[3]).

which means that

light

cone in

parameters to2,

degenerate

subspace

ut12 ig eele

lmo2(q-ll)2

for,

q.i)ll

O

for

lq.,l<}l==sVii3,

}mev2<q+lt}'

for

q.i<-lt

l=Vi/S).

If

V:=O,

we obtain

oseillators with

the

same m

(2di)-t

-(2d,a)'tat O O -(2d,)-T2ai

(2d,2)uZ+ai2(2diS)-i

O

O '

O

O

(2d2)-i

-(a2)(2d22)-i

O O -a,(2ck2)ni

_{(2d22)-i+ai!(2d,")-i}

'

g=16-'(a,d2)-S;O

for

appropriate

domain

of

_parameters.

'

into

two 2-dimensional spaees

(j'=1,2)

with the curvatures:

RSj2)n=

J

(j

ij

i

sS

1

J

(8cl

4)-i _a -- _{a2-- a +- a`)}

_(J'=-1,

₂₎ ,

Kq)=(4ci,)-i(aj--2- at2--2- ats+t a")==-}R(i)

(o'=i,2)

.

is

the _quantum case which Ieannot

discuss

here in

detail

(I

refer tothe

I

would

like

only to say that in all exaetly caleulable cases one _{obtains g=O}

the spaee

is

a vector

bundle

in _which in some

direetion

dsZ=::O

as on the the spaee-time.

In

the ease of a

2-dimensional

oseillator

described

by

3

to!, T we obtain _g=O,

but

rank(g..)==2 and so we _get a

Zdimensional

with some curvature.

Further

phygieal･and geQmetrical

discussion

.is

information 71leermodynamicsand

Diferentiat

Geometry

given

in

[3].

As

regards

the

higher

order temperatures, these cases cannot

be

_exactly ¢

aleu-lated

without

introdueing

new special functions which as _yet are enly partially

investigated.

References

[1]

R.S. Ingarden,Differential geometry and _physics,tTlensor30

(1976),

201-209.

[2]

Y. Sato,K. Sugawa and M. Kawaguchi, A geometrical strueture of the _parameter space of the

two-dimensional normal distributions,to be _published inRep. Math. Phys.

[3]

R.S. Ingarden,Y.

Sato

and

M.

Kawaguchi,

The

information

_geometry of the control space and

its

applicatien to thermodynamies, tebe published inRep. Math. Phys.

[4

]

R.S.Ingardenand A.Kossakowski, On the eonnection ofnon-equilibrium informationthermodynamics

with non-Hamiltonian quanturn rnechanies of open systems, Ann. Phys.

(New

York) 89

(1975),

485.

[5]

S.Kullbaek and R.A. Leibler,Informationand suMcieney, Ann. Math. Stat.22

(1950),

79-86.

[6]

S.Kullback,Information theory and statistics, Wiley, New York 1959.

[7]

A. Ronyi,Probability theory, Akad. Kiado, Budapest 1970.

[8]

F. Sch16gl,Zur statistischen Theorie der Entropieproductionin nichtabgeschlossenen Systemen,

Z.

Phys. 191

(1966)

81-eo.

[9]

R,A. Fisher,Statisticalutilization of multiple measurements, Ann. Eugenics8

(1938),

376-386,also

in: Contributionsto mathematical statisties, Wiley, New York, 1950

(by

R.A. Fisher).

[10]

C.R.

Rao,

Information

and theaeeuracy attainable inthe estimation of statistieal _problems, Bull,

CalcuttaMath. Soc.37

(1945),

81-91.

[11]

P.O. Mahalanobis,On the generalized distaneein statistics, Proc. Nat. Inst. Sci.of India2

(1936),

No. 1.