Fluctuations and Brownian Motion 2  fluorescent spheres in water (left) and DNA solution (right) (Movie Courtesy Professor Eric Weeks, Emory University: Brownian motion in water Brownian motion of DNA Copyright (c) Stuart Lindsay 2008

Fluctuations One set of collisions5·10 5 set of collisions Simulated distribution of speeds in a population of 23 atoms of an ideal gas: V=1.25·10 3 nm 3 (50·50·50nm); T=300K.

Calculating Fluctuations Taking the second derivative of dlnZ/d  :

This is just the mean square thermal average of E:

For an ideal gas: The relative size of energy fluctuations scales as

In an open system:

From thermodynamics: Isothermal compressibility For an ideal gas:

The result just obtained for energy holds for all quantities (extensive quantities) that, like energy, grow with N (T, E, P, V, S). In general the root-mean-square value of the fluctuations relative to the mean value of a quantity is given byN Relative fluctuation % % Copyright (c) Stuart Lindsay 2008

Brownian motion Langevin equation: α (friction coefficient ) For a sphere of radius a in a medium of viscosity η:  =6πηa Stoke’s force Stoke’s force = friction exerted on the particle by the fluid. For small velocities, it is proportional to the velocity v.

F(t) is a random force representing the constant molecular bombardment exerted by the surrounding fluid: Average is zero!Finite only over duration of single “effective” collision F(t) is independent of the velocity of the particle (v) and varies extremely rapidly compared to the variations in v. There is no correlation between F(t) and F(t+Δt) even though Δt is expected to be very small.

Multiplying both sides of the Langevin eqn. by x and using Re-arranging and taking thermal averages: = 0

so substituting yields To find A, note

with =0 at t=0: substitute Integrate with =0 at t=0:

Long time solution: Mean square displacement increases with t! Copyright (c) Stuart Lindsay 2008

Now we see why the sphere in a viscous DNA solution moves more slowly! 2  fluorescent spheres in water (left) and DNA solution (right) Copyright (c) Stuart Lindsay 2008 (Movie Courtesy Professor Eric Weeks, Emory University:

The Diffusion Equation Flow of solute or heat under action of random forces: J is flux per second per unit area The flux is the change in concentration across a surface multiplied by the speed with which particles arrive:

Using: From which Diffusioncoefficient Fick’s first law Fick’s second law Diffusion Equation In 3D:

The solution in 1-D for a solute initially added as a point source is: A=1 As t→ ∞, the distribution becomes uniform, the point of ‘half-maximum concentration’ x ½ advancing with time according to:

Einstein-Smoluchowski Relation The mobility is defined as the inverse of the friction coefficient: Einstein-Smoluchowski relation Compareto A surprising relation between thermal motion and driven motion: the diffusion constant is the ratio of kT to the friction constant! A fundamental relation between energy dissipation and diffusion

Viscosity is not an equilibrium property, because viscous forces are generated only by movement that transfers energy from one part of the system to another. Ex. Motion of a spherical large particle with respect to a large number of small molecules. Force radius viscosity speed Stokes’ law The introduction of bulk viscosity requires that the small molecules rearrange themselves on very short times compared with the time scale of the motion of the sphere.

Diffusion, fluctuations and chemical reactions 1.If the reactants are not already mixed they need to come together by diffusion. 2.Once together, they need to be jiggled by thermal fluctuations into a “transition state” 3.If the free energy of the products is lower than that of the reactants the products lose heat to the environment to form stable end products.

Haber process for Ammonia The transition state Eyring transition state theory Copyright (c) Stuart Lindsay 2008

The entropy adds a temperature-dependent component to the energy differences that determine the final equilibrium state of a system.

Kramers’ Theory of Chemical Reactions Noise driven escape: Thermal fluctuations allow the particles in the well to rapidly equilibrate with the surroundings. The motion of the particles over the barrier is much slower.

The Kramers Model Reaction coordinate Microscopic description of the prefactor in terms of potential curvature

Unimolecular reactions Not very common, but can be described as a one step process by the Kramers theory Example is isomerism of isonitrile Copyright (c) Stuart Lindsay 2008

Thermodynamic Potentials for Nanosystems The Gibbs free energy in a multicomponent system is: This equation contains no reference to system size (all quantities are extensive: doubling the volume of a system, doubles its free energy) Nanosystems at equilibrium derive their “stable” size from surface and interface effects which are not extensive.Nanosystems at equilibrium derive their “stable” size from surface and interface effects which are not extensive. Ex. Self-assembly originates from a competition between bulk and surface energies of the phases that self-assemble (stability of colloidal systems). Copyright (c) Stuart Lindsay 2008

Hill has generalized thermodynamics to include a “subdivision potential” The simplest approach add surface terms to the free energy. Subdivision potential number of independent parts of the system

Modeling nanosystems explicitly: Molecular Dynamics Limited to  s timescales by required time step/processor speed. Interatomic vibrations are rapid (≈10 13 Hz) so that the time step in the calculations must be much smaller (≈ s). Copyright (c) Stuart Lindsay 2008

The number of calculations scales as N 2 (it depends on the problem). If all possible pair interactions need to be considered, the number of interactions would scale like 2 N. Limitations of MD Timescale is limited to ns or  s at best. This is fine for harmonic vibrations but nanosystems like enzymes work on ms timescales. Ex. Viscously damped motion: Ex. Activated transitions over barriers:

Example of MD Cyclic Sugar molecule being pulled over DNA Qamar et al. Biophys J. (2008) 94, 1233 Movie link: 13mer.avi

Thermal Fluctuations and Quantum Mechanics The density matrix formulation of quantum mechanics allows description of the time evolution of a system subject to time dependent forces. Copyright (c) Stuart Lindsay 2008

Thermal fluctuations can be represented by the “random phase approximation” – quantum interference effects are destroyed.

Uncertainty Principle and fluctuations Interaction energy, , will decay exponentially at large distances When  kT The time characteristic of a molecular vibration. Copyright (c) Stuart Lindsay 2008