Introduction to Statistical Thermodynamics

of Soft and Biological MatterLecture 3

Statistical thermodynamics III

• Kinetic interpretation of the Boltzmann distribution. Barrier crossing.• Unfolding of single RNA molecule.• Diffusion. • Random walks and conformations of polymer molecules.• Depletion force.

Boltzmann distribution

• System with many possible states (M possible states) (different conformations of protein molecule) Each state has probability Each state has energy

Partition function:

Maxwell-Boltzmann distribution

Probability distribution for velocities:

Gas of N molecules: - velocity of a molecule

How to compute average…

If you want to derive the formula yourself…

Use the following help:

Example: fluctuations of polymer molecule

verify:- energy of polymer molecule

Probability distribution:

Equipartition theorem:

Example: Two state system

Probability of state:

Verify!

Kinetic interpretation of the Boltzmann distribution

- activation barrier

Reaction rates:

Kinetic interpretation of the Boltzmann distribution

- activation barrier

Detailed balance (at equilibrium):

Number of molecules in state 2 and in state 1

Verify!

Unfolding of single RNA molecule

J. Liphardt et al., Science 292, 733 (2001)

Optical tweezers apparatus:

J. Liphardt et al., Science 292, 733 (2001)

Two-state system and unfolding of single RNA molecule

Extension

Open state: Close state (force applied):

force extension

Diffusion

Robert Brown: 1828

Albert Einstein

Pollen grain (1000 nm)

Water molecules (0.3 nm):

Universal properties of random walk

0

L (step-size of random walk)

- random number (determines direction of i-th step)

One-dimensional random walk:

N-th step of random walk:

(N-1)-th step of random walk:

Verify!

Diffusion coefficient

From dimensional analysis:

Number of random steps N corresponds to time t:

Friction coefficient:

Diffusion coefficient and dissipation

Viscosity Particle size

Einstein relation:

Diffusion in two and three dimensions

One-dimensional (1D) random walk:

Two-dimensional (2D) random walk:

Three-dimensional (3D) random walk:

Conformations of polymer molecules

* Excluded volume effects and interactions may change law!

L – length of elementary segment

• Universal properties of random walk describe conformations of polymer molecules.

More about diffusion… Diffusion equation

Surface area: A

x

Flux:

c – concentration of particles

Solution of diffusion equation

verify this is the solution!

c(x,t)

x

The concentration profile spreads out with time

Pressure of ideal gasFree energy of ideal gas:

density:N – number of particlesV - volume

Pressure:

Osmotic forces: Concentration difference inducesosmotic pressure

Semi-permeable membrane(only solvent can penetrate)

Protein solution

Depletion force

R