In a state of dynamic equilibrium, and under the hypothesis of isothermal fluid, the particles are distributed according to the barometric distribution
where ρ − ρo is the difference in density of particles separated by a height difference, of
h
=
z
z
o
{\displaystyle h=z-z_{o}}
, kB is the Boltzmann constant (the ratio of the universal gas constant, R, to the Avogadro constant, NA), and T is the absolute temperature. , the probability density of the particle incrementing its position from
x
{\displaystyle x}
to
x
description +
{\displaystyle x+\Delta }
in the time interval
{\displaystyle \tau }
). The approximation is valid on short timescales.
Brownian motion, or pedesis (from Ancient Greek: πήδησις /pɛ̌ːdɛːsis/ “leaping”), is the random motion of particles suspended in a medium (a liquid or a gas). On small timescales, inertial effects are prevalent in the Langevin equation.
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This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem. The multiplicity is then simply given by:
and the total number of possible states is given by 2N.
The above solution
S
t
{\displaystyle S_{t}}
(for any value of t) is a log-normally distributed random variable with expected value and variance given by2
They can be derived using the fact that
Z
t
=
exp
(
W
t
1
2
2
t
)
{\displaystyle Z_{t}=\exp \left(\sigma W_{t}-{\frac {1}{2}}\sigma ^{2}t\right)}
is a martingale, and that
The probability density function of
S
t
{\displaystyle S_{t}}
is:
To derive the probability density function for GBM, we must use the Fokker-Planck equation to evaluate the time evolution of the PDF:
where
(
S
)
{\displaystyle \delta (S)}
is the Dirac delta function. .