412
9 Bioenergetics
9.23.2 Flows Near Equilibrium
A large set of interacting particles, acting classically, such as atoms and molecules
in your body, if allowed to relax, will tend to distribute in energy states which follow
the Boltzmann distribution, simply because this distribution contains a large number
of microstates for a given macrostate and therefore this distribution is the most
likely. 69
In moving toward equilibrium, or under continuous external influences, flows of
energy, entropy, material and charges can occur. Internally, these flows are caused
by gradients of intrinsic quantities, such as temperature, concentrations, and electric
potentials, or by chemical imbalances.
The time rate of change of any one of the extensive variable densities Y i /N ≡ ρ i
in a small subsystem with fixed total particle number N and located near {x}, will
be denoted ˙
ρ i . These rates can depend on position and time through the variations in
the local values of the Y k : ˙
ρ i = F i (Y (x, t)). At equilibrium, as well as steady-state
conditions, there should be no more changes in time. It follows that a Taylor series
of ˙
ρ i to first order in δY j ≡ Y j −
Y j
about equilibrium values should be of the
form
˙
ρ i =
k
∂F i
∂Y k
δY k ≡ −
k
α ik δY k ,
(9.247)
where the α ik are constants. A minus sign is introduced anticipating that the changes
with time will exponentially drop with time.
Similarly, the changes in the conjugate thermodynamic variables y l =
−(1/k B )∂S/∂Y l can be expanded around the equilibrium values of Y k to give
δy l =
m
γ lm δY m .
(9.248)
By defining λ ij =
l α il γ
−1
lj , the last two equations combine to give
˙
ρ i = −
j
λ ij δy j .
(9.249)
For simple systems y ε = −(1/k B T ), y v = −(p/k B T ), and y n = (μ/k B T ),
and we let ε ≡ E/N. The rates of change with time of the density of extensive
variables in each small subsystem of fixed total particle number are given in terms of
changes from equilibrium in the intensive quantities, such as temperature, pressure,
and chemical potentials.
69 At or near equilibrium, bosons and fermions densely populating the lowest energy states will
deviate from the Boltzmann distribution.
9 Bioenergetics
9.23.2 Flows Near Equilibrium
A large set of interacting particles, acting classically, such as atoms and molecules
in your body, if allowed to relax, will tend to distribute in energy states which follow
the Boltzmann distribution, simply because this distribution contains a large number
of microstates for a given macrostate and therefore this distribution is the most
likely. 69
In moving toward equilibrium, or under continuous external influences, flows of
energy, entropy, material and charges can occur. Internally, these flows are caused
by gradients of intrinsic quantities, such as temperature, concentrations, and electric
potentials, or by chemical imbalances.
The time rate of change of any one of the extensive variable densities Y i /N ≡ ρ i
in a small subsystem with fixed total particle number N and located near {x}, will
be denoted ˙
ρ i . These rates can depend on position and time through the variations in
the local values of the Y k : ˙
ρ i = F i (Y (x, t)). At equilibrium, as well as steady-state
conditions, there should be no more changes in time. It follows that a Taylor series
of ˙
ρ i to first order in δY j ≡ Y j −
Y j
about equilibrium values should be of the
form
˙
ρ i =
k
∂F i
∂Y k
δY k ≡ −
k
α ik δY k ,
(9.247)
where the α ik are constants. A minus sign is introduced anticipating that the changes
with time will exponentially drop with time.
Similarly, the changes in the conjugate thermodynamic variables y l =
−(1/k B )∂S/∂Y l can be expanded around the equilibrium values of Y k to give
δy l =
m
γ lm δY m .
(9.248)
By defining λ ij =
l α il γ
−1
lj , the last two equations combine to give
˙
ρ i = −
j
λ ij δy j .
(9.249)
For simple systems y ε = −(1/k B T ), y v = −(p/k B T ), and y n = (μ/k B T ),
and we let ε ≡ E/N. The rates of change with time of the density of extensive
variables in each small subsystem of fixed total particle number are given in terms of
changes from equilibrium in the intensive quantities, such as temperature, pressure,
and chemical potentials.
69 At or near equilibrium, bosons and fermions densely populating the lowest energy states will
deviate from the Boltzmann distribution.