208

DOC.

5

LOCALIZATION OF ELECTROMAGNETIC ENERGY

this

latter view

should be

adopted.2

The

considerations

on

which I

based

myself

rest

on

a

principle

of

Boltzmann's,

according

to which

the

entropy

S

and the

statistical

probability

W

of

a

state

of

an

isolated

system

are

connected

by

the relation

S

=

-log

W,

N

s

where

R

is

the

gas

constant

for

a

perfect

gas

and

N

the number

of

molecules in

one

gram-molecule.

If

a

complete

molecular

picture

of the

system

considered

is

given, one

can

calculate

the

statistical

probability

W

for

each state

of the

system,

and

from this

one

can

calculate S with

the

aid

of the

formula.

If,

conversely,

the

system

is

known

thermodynamically,

then

one

will know

S,

and from this

one

will be able to derive

the

statistical

probability

of each

state

of the

system.

To be

sure, one

cannot establish

an

elementary theory

(e.g.,

a

molecular

theory)

of the

system

from

W in

a

unique

and

well-defined

fashion; but, still, any

theory

giving

the

wrong

values

of

W

for

any

of the

states

can

be

considered

unacceptable.

One

can

then

find

the

entropy

of radiation

in

empty

space by means

of

thermodynamics,

using

the

law

of

black-body

radiation,

and

solve

the

following

problem:

consider

two

spaces

enclosed

within

impermeable

walls

and

connected

by a

tube that

can

be

closed;

let

V

be the

volume

of

one

of the

spaces

and

V0

the total

volume;

assume

that these

spaces are

filled with

a

radiation whose

frequency

lies

between

v

and

v +

dv,

and

whose

total

energy

is

E0.

We seek

to calculate

the

entropy

S

of the

system

for

every possible

distribution of the

energy

E0

between the

two

spaces.

From the

entropy S

of

each

of these

possible

distributions,

one can

deduce

the

statistical

probability corresponding

to each

of

them.

In

this

way one

finds for

a

sufficiently

dilute

radiation the

following expression

for the

probability

that

at

a given

moment all

of the

energy

E0

is

contained

in

the volume

V:

W

=

V

K

\e,

[7]

It

can easily

be

shown

that

this

expression

is not compatible

with

the

principle

of

superposition.

As

regards

the distribution between the

two

spaces,

the radiation

behaves

as

if its

energy

were

localized in

E0/hv

points

moving

independently

of each other.

From

this it

follows-unless

one

wants to

admit that the

use

of

impermeable

walls in

these considerations

is inadmissible-that,

regarding

the

localization

of

its

energy,

the

radiation

must

itself

have

a

structure not

given by

the

ordinary

theory.

In

conclusion,

let

me say

that the

commonly

accepted

localization

of

energy

(just

like

the

momentum in

the

electromagnetic

field)

is

by no means a necessary

consequence

of

the

Maxwell-Lorentz

equations.

Furthermore,

one

can

give,

for

example,

a

distribution

compatible

with

the mentioned

equations that,

for

static and

stationary states,

coincides

[8]

completely

with

the

one

given by

the old

theory

of

action

at

a

distance.

[6]

2A.

Einstein,

Ann.

d.

Phys.,

4,

17

(1905):

139ff.