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 Two degrees of freedom for storing energy: kinetic and elastic; l There is a continuous flow of energy between the two dofs; l 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 The diffusion coefficient a spherical particle of D of any object subject radius R is a liquid of to thermal agitation at viscosity η is 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 T D = B 6πηR Jean Perrin and the size of atoms http://www.nobelprize.org/nobel_prizes/physics/laureates/1926/press.html Dynamic Light Scattering (DLS) sample Incident beam Laser * Speckle pattern * Courtesy of Oleg Shpyrko (http://oleg.ucsd.edu/) The physical origin of Speckle ∑sin(φ)≈ 0 ∑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. Intensity correlation function sample Incident beam Laser Correlator Intensity time The correlator calculates the correlation function of the measured light intensity ) I(t): τ G( I(t)I(t +τ ) G(τ ) = τ I(t) 2 Analyzing DLS data in terms of diffusion coefficients The characteristic time for intensity fluctuation is roughly the time needed for a particle to diffuse over a distance comparable to the Intensity wavelength time τ ≈ λ2 / D 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 4πn ⎛θ ⎞ g(τ )=1+ exp(− 2Dq 2τ ) with 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 alpha- crystallin protein structures follows from a « diabetogenic diet » http://www.grc.nasa.gov/WWW/RT/RT1999/6000/6712ansari.html Back to Boltzmann’s law 1 0.8 0.6 Prob 0.4 300 K 0.2 5000 K 0 106 K −4 −2 0 2 4 10 10 10 10 10 E (eV ) 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 .