CHIM0698
Thermal Fluctuations
2014-2015
What is thermal equilibrium?
l
http://falstad.com/gas/
Energy spreads evenly in all available degrees of freedom.
In this case: all molecules and all three components of the velocity
Diatomic molecules
l
http://webphysics.davidson.edu/
physlet_resources/thermo_paper/thermo/
examples/ex20_4.html
One additional degree of freedom here: the rotation.
The simulation shows that:
1)  the rotation has the same energy as one component of the
translation.
2)  The translation energy of the monoatomic and diatomic
molecules is the same (i.e. mass is irrelevant)
Each degree of freedom has the same average energy
Yet another example: molecules of
different sizes
Mass-spring analogy
l
l
l
Two degrees of freedom for storing energy:
kinetic and elastic;
There is a continuous flow of energy between
the two dofs;
The total energy is evenly shared between
the two dofs.
A remark
Note that the equilibration of a warm and cold
media put in contact is a particular case of this
more general rule.
Initially, only the dofs of the warm medium did
store some energy;
Eventually, the dofs of both the warm and cold
bodies
Equiprobability
The entire field of thermodynamics seem to lie
in this simple statement:
At thermodynamic equilibrium, all the
microstates of an isolated system are equally
probable.
Boltzmann’s formula
Ludwig Boltzmann 1844 - 1906
Number of microstates corresponding to a given total energy E
(very little to do with “disorder” actually)
In contact with a reservoir at
temperature T
Reservoir
Temperature T
System
Energy exchange
A macroscopic example
From the
Feynman Lectures on Physics
A mesoscopic example: Brownian
motion
l
http://www.youtube.com/watch?
v=cDcprgWiQEY
The giant beach ball analogy to
Brownian motion
The Brownian particle
The invisible molecules
pushing randomly
Size-dependence of diffusion coefficients:
the Stokes-Einstein relation
The mobility µ = v/F of
a spherical particle of
radius R is a liquid of
viscosity η is
The diffusion coefficient
D of any object subject
to thermal agitation at
temperature T is
µ = 1 /(6πηR)
D = µ kBT
The diffusion coefficient D of a spherical particle of radius R, in a
liquid with viscosity η , at temperature T is given by
k BT
D=
6πηR
Jean Perrin
and the size of atoms
http://www.nobelprize.org/nobel_prizes/physics/laureates/1926/press.html
Dynamic Light Scattering (DLS)
sample
Laser
Incident beam
*
Speckle pattern
* Courtesy of
Oleg Shpyrko
(http://oleg.ucsd.edu/)
The physical origin of Speckle
∑ sin(φ ) ≈ 0
When the particles are hit by electromagnetic wave,
they reemit it in all directions (Rayleigh scattering).
The speckle pattern results from the interference of
these secondary reemitted waves.
∑ sin(φ ) >>0
Intensity correlation function
sample
Laser
Correlator
Intensity
Incident beam
The correlator calculates the correlation
function of the measured light intensity
I(t):
G (τ ) =
I (t ) I (t + τ )
I (t )
2
G(τ)
time
τ
Analyzing DLS data in terms of
diffusion coefficients
Intensity
The characteristic time for intensity fluctuation
is roughly the time needed for a particle to
diffuse over a distance comparable to the
wavelength
2
τ ≈λ /D
time
Using λ = 633 nm (He-Ne laser) and D = 10-10 m²/s (5 nm particle in water
at room temperature), one finds τ = 4 ms.
More accurate analysis leads to
(
2
g (τ ) = 1 + exp − 2 Dq τ
)
with
4πn
⎛ θ ⎞
q=
sin⎜ ⎟
λ0
⎝ 2 ⎠
Example 1: Fullerene colloidal dispersions
Example 2: an in-situ biological application
Early diagnosis of diabetes with
DLS analysis of lenses
An increase in size of alphacrystallin protein structures
follows from a « diabetogenic
diet »
http://www.grc.nasa.gov/WWW/RT/RT1999/6000/6712ansari.html
Back to Boltzmann’s law
1
P rob
0.8
0.6
0.4
300 K
0.2
5000 K
0
−4
−2
10
10
0
10
2
10
E (eV )
4
10
106 K
Average quantities
Equipartition theorem
Every degree of freedom in a system carries an
average energy 1/2kBT
You may know this as:
“every degree of freedom adds R/2 to the heat
capacity”
Fluctuations
Proportional to square root the the size
Link to thermodynamics
Brownian ratchets
… or how nanomachines cannot work
Carnot’s principle, here again
Cell motility results from nm-scale
force generation
https://www.youtube.com/watch?v=I_xh-bkiv_c
https://www.youtube.com/watch?v=ebkRySBa3JU
Actin
polymerization
is a very
general
process
Nature Reviews Molecular Cell Biology 7, 404-414 (June 2006) | doi:10.1038/nrm1940
Actin polymerization: actual
mechanism
The gist of it: nanomachines
harvest thermal fluctuations
Nature Reviews Molecular Cell Biology 7, 404-414 (June 2006) | doi:10.1038/nrm1940
The force-velocity relation
follows Boltzman’s law
Brangbour C, du Roure O, Helfer E, De ́moulin D, Mazurier A, et al. (2011)
Force-Velocity Measurements of a Few Growing Actin Filaments.
PLoS Biol 9(4): e1000613. doi:10.1371/journal.pbio.1000613