1. wwilson@uco.edu Thermodynamics of a Single Particle System W. J. Wilson Department of Engineering and Physics University of Central Oklahoma Edmond, OK 73034 2011 Oklahoma Academy of Sciences AnnualTechnical Meeting Southeastern Oklahoma State University Durant, OK November 11, 2011

2. wwilson@uco.edu Thermodynamics of a Particle? Single particle thermodynamics is a bit of an uncharted subject. Some, steeped in statistical basis of thermodynamics, conclude that the laws of thermodynamics fall apart at the single particle level. “The law of entropy increasing is only a statistical law; it is not ‘fundamental’ because it cannot describe the behavior of an individual atom or molecule; it deals with the average number of them. Entropy is not a concept that can be meaningfully applied to a single particle, or even to a small number of particles.” – John Wheeler

3. wwilson@uco.edu Thermodynamics of a Single Particle System

4. wwilson@uco.edu First Law – Sign Convention The positive work done by system sign convention (used in physics and engineering) is

5. wwilson@uco.edu The positive work on system sign convention (used in chemistry and some upper-level physics treatments) is

6. wwilson@uco.edu 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,

7. wwilson@uco.edu Now if the change dx and dv occur in time dt So for a single particle system,

8. wwilson@uco.edu 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

9. wwilson@uco.edu 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

10. wwilson@uco.edu Integrating Factor For a single particle, U = KE+PE = U(x,v) and F = F(x,v) are functions of position and velocity only so

11. wwilson@uco.edu For “adiabatic” processes or The right-hand side is a function of x and v only, so we are guaranteed a unique solution through a given initial state (x 0,v 0 )

12. wwilson@uco.edu For “adiabatic” processes

13. wwilson@uco.edu For “adiabatic” processes and So we must have a factor λ(x,v) such that or

14. wwilson@uco.edu Now since These yield an exact differential As So

15. wwilson@uco.edu Summary / Conclusions 15 The natural sign convention for physics There is an exact differential (“entropy”) for the single particle References 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).