Entropy and the Thermodynamics of a Single Particle W. J. Wilson Department of Engineering and Physics University of Central Oklahoma Edmond, OK Oklahoma Academy of Sciences Annual Technical Meeting University of Central Oklahoma Edmond, OK November 9, 2012

First Law – Sign Convention Chemistry and Physics sign convention

For a single particle, U = KE+PE = U(x,v) and F = F(x,v) are functions of position and velocity only so if we use the work on system positive sign convention which becomes for a free(non-relativistic) particle,

Now if the change dx and dv occur in time dt So for a single particle system,

So for a single particle system, And for a closed system (no energy allowed to enter or escape) (i.e., an “Adiabatic” Process) We find

So for a single particle system, using the work done on the system is positive sign convention, 1.Yields Newton’s 2 nd Law in the standard form 2.Particle motions governed by Newton’s 2 nd Law correspond to “adiabatic” processes

Generalized Newton’s 2 nd Law For a single particle, or

Now if the change dx and dv occur in time dt So for a single particle, isothermal system

So, Or Generalized 2 nd Law !

The extra “Force-Like” term in is an entropic force or

Entropic Force Example Long Chain Molecules

Expansion of Universe is Accelerating 2011 Physics Nobel Prize Saul Perlmutter (Lawrence Berkeley Lab) Brian Schmidt (Australian National Lab) Adam Reiss (Johns Hopkins)

Dark Energy as Entropic Force? Accelerating expansion

Summary 1 st Law of Thermodynamics (TdS = dU – Fdx ) applied to a single particle leads to a generalized 2 nd Law The entropic force term could give an explanation of the observed accelerating expansion of the universe.

15 References W. J. Wilson, The Thermodynamics of a Single Particle System, 2012 Oklahoma Academy of Sciences Annual Technical Meeting, Durant, OK, November 11, 2011.Slides available at E.C.G. Stueckelberg and P. B. Scheurer, Thermocinéque Phénoménologique Galiléenne, (Birkhauser, Stuttgart, 1974) P.E. Williams, On a Possible Formulation of Particle Dynamics in Terms of Thermodynamics Conceptualizations and the Role of Entropy in It (M.S. Thesis, Naval Postgraduate School, Monterey, CA, 1976).