214
BROWNIAN MOTION
Vt2
=
2R
N
(II)
where A
is
the
change
in the
parameter
a,
and
B
is
what
Einstein calls the
mobility
of
the
system
with
respect
to
a. By considering
the effect
on
the motion
of
dissipation
alone
as
a
function
of
time,
Einstein
was
able to estimate the time interval below which his
results
are no
longer valid,
due to the breakdown
of
the assumed
independence
of
events in
successive time
intervals.[55]
V
Einstein's
further studies
of
fluctuation
phenomena
elaborate the two
fundamental
approaches
in his first two
papers on
Brownian motion: the
thermodynamical
approach
to
fluctuations that
emerged
from his work
on
statistical
mechanics,
and
the
stochastic treat-
ment
of
fluctuations that takes
dissipation
and time
dependence
into
account.[56]
By con-
centrating on a
few basic features
of
statistical
physics,
Einstein
further reduced
the need
for
detailed
microphysical assumptions
and
simplified
the technical
aspects
of
his
analysis.
All
of
these
subsequent
studies
emphasize applications
of
his
methods;
by
1909,
applica-
tions to
black-body
radiation
were again
of
primary concern
to
Einstein.[57]
Einstein further
developed
the
thermodynamical approach
in
a
paper
on voltage
fluctua-
tions in
a
condenser
(Einstein
1907b
[Doc. 39]).
He
gave a
simple
derivation of
a
formula
for
mean square fluctuations, a
derivation that does not
depend on dynamical premises,
but is
directly
based
on
Boltzmann's
principle relating probability
and
entropy, both
con-
ceived
by
Einstein
as thermodynamical quantities. Consequently,
he stated
that
his treat-
ment
of
fluctuations does not
require any "definite
stipulations
concerning
the
molecular
model
to
be
applied" ("bestimmte
Festsetzungen
in
betreff
des anzuwendenden moleku-
laren
Bildes").[58]
The
new
element in
Einstein's
argument
is
his definition
of
probability
in
Boltzmann's
principle as
the
"statistical
probability
of
a
state"
("statistische
Wahrscheinlichkeit eines Zustandes"),
a
concept
introduced
in
Einstein 1905i
(Doc.
14).[59]
Einstein further elaborated this
concept
in
Einstein 1909b
(Doc. 56),
where he
applied
the
mean square
fluctuation formula derived from
Boltzmann's
principle
to
energy
fluctuations in
black-body
radiation.[60]
[55]
For
a
discussion
of
subsequent
work
on
the
significance
of
such
a
time interval in the
analy-
sis
of
Brownian motion, see
Fürth
1922,
pp.
60-61, fn. 8.
[56]
For studies
of Einstein's
thermodynamical
approach
to
fluctuations, see
Klein
1967, 1974b,
1982a. For
a
review
of Einstein's
work on
the
stochastic
approach,
see
Sciama 1979.
[57]
See
Einstein
1909b
(Doc.
56);
for
a
dis-
cussion
of Einstein's
work
on black-body ra-
diation fluctuations,
see
the editorial
note,
"Einstein's
Early
Work
on
the
Quantum Hy-
pothesis," p.
146.
[58]
Einstein 1907b
(Doc.
39),
p.
569.
[59]
For
a
discussion
of the
concept
of
proba-
bility developed
in Doc. 14, see
the editorial
note,
"Einstein's
Early
Work
on
the
Quantum
Hypothesis,"
p.
138.
[60]
For further discussion
of this
paper,
see
the
editorial note,
"Einstein's
Early
Work
on
the
Quantum Hypothesis," pp.
145-146.
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