10.10 Fluctuations
499
ways of distributing the energy in the system among the possible energy states.
When a system is far from equilibrium, the calculation of S is more difficult,
particularly for systems which change rapidly in time, such as one which explodes.
But as we have seen, systems such as tissue and fluids in live biological cells, in
which small subsystems are not far from equilibrium, can be tractable. The small
subsystems must be large enough that fluctuations in thermodynamic variables are
small compared to their values.
In investigating thermodynamic fluctuations, Einstein recognized that Boltzmann’s formula could be put in a form appropriate for calculating the average of
fluctuations in the thermodynamic variables, provided the system is not too far from
equilibrium. 61
Einstein expressed Boltzmann’s relation as
W = exp (S(Y 1 , Y 2 , · · · , Y n )/k B ) ,
(10.222)
where the set {Y 1 , Y 2 , · · · , Y n } represent the full set of n extrinsic thermodynamic
variables, such as local values of internal energy, volume, and number of various
particle species. The W , which we identified as the number of ways particles can
be distributed in given energy states, is also proportional to the probability that the
system would have the given entropy S for that particle distribution. 62
Each of the thermodynamic variables Y i could depend on position in the system
and on the time of measurement. If so, the regions around a given position r will
be considered a thermodynamic subsystem when it contains enough particles and
energy states to make fluctuations in the thermodynamic variables in the given
region small compared to their values after a characteristic relaxation time for the
system, i.e. the time for the system to reach equilibrium or a steady state if there
were no external influences.
Note that, as we observed with Eq. (9.76), because the entropy is an extensive
quantity, we have
S(λY ) = λS(Y )
(10.223)
so that the Euler relation
S =
i
∂S
∂Y i
Y i
(10.224)
61 A. Einstein, Über die Gültigkeitsgrenze des Satzes vom thermodynamischen Gleichgewicht und
über die Möglichkeit einer neuen Bestimmung der Elementarquanta, Annalen der Physik 22 569–
572 (1907). These days, fluctuations in thermodynamic variables are commonly calculated using a
partition function, but Einstein’s method is an elegant and simple alternative.
62 Einstein remarked that if he had seen Willard Gibbs’ work on statistical mechanics, he would
have not written his 1902–1905 papers on the subject. Our discussion here is based on Einstein’s
development, as it reflects Einstein’s intuition, simplicity and his ability to focus on the essence of
a problem.
499
ways of distributing the energy in the system among the possible energy states.
When a system is far from equilibrium, the calculation of S is more difficult,
particularly for systems which change rapidly in time, such as one which explodes.
But as we have seen, systems such as tissue and fluids in live biological cells, in
which small subsystems are not far from equilibrium, can be tractable. The small
subsystems must be large enough that fluctuations in thermodynamic variables are
small compared to their values.
In investigating thermodynamic fluctuations, Einstein recognized that Boltzmann’s formula could be put in a form appropriate for calculating the average of
fluctuations in the thermodynamic variables, provided the system is not too far from
equilibrium. 61
Einstein expressed Boltzmann’s relation as
W = exp (S(Y 1 , Y 2 , · · · , Y n )/k B ) ,
(10.222)
where the set {Y 1 , Y 2 , · · · , Y n } represent the full set of n extrinsic thermodynamic
variables, such as local values of internal energy, volume, and number of various
particle species. The W , which we identified as the number of ways particles can
be distributed in given energy states, is also proportional to the probability that the
system would have the given entropy S for that particle distribution. 62
Each of the thermodynamic variables Y i could depend on position in the system
and on the time of measurement. If so, the regions around a given position r will
be considered a thermodynamic subsystem when it contains enough particles and
energy states to make fluctuations in the thermodynamic variables in the given
region small compared to their values after a characteristic relaxation time for the
system, i.e. the time for the system to reach equilibrium or a steady state if there
were no external influences.
Note that, as we observed with Eq. (9.76), because the entropy is an extensive
quantity, we have
S(λY ) = λS(Y )
(10.223)
so that the Euler relation
S =
i
∂S
∂Y i
Y i
(10.224)
61 A. Einstein, Über die Gültigkeitsgrenze des Satzes vom thermodynamischen Gleichgewicht und
über die Möglichkeit einer neuen Bestimmung der Elementarquanta, Annalen der Physik 22 569–
572 (1907). These days, fluctuations in thermodynamic variables are commonly calculated using a
partition function, but Einstein’s method is an elegant and simple alternative.
62 Einstein remarked that if he had seen Willard Gibbs’ work on statistical mechanics, he would
have not written his 1902–1905 papers on the subject. Our discussion here is based on Einstein’s
development, as it reflects Einstein’s intuition, simplicity and his ability to focus on the essence of
a problem.