A Schrödinger-like equation for the Thermodynamics of a particle in a box

Abstract

The particle in an expanding/contracting 1-dimension box is revisited in action-angle like variables with direct thermodynamic interpretation. An angle dependent potential is proposed accurately describing the mechanical behavior while also capturing thermodynamic evolution -- entropy production -- within a canonical Hamiltonian framework. Heat transfer at constant volume is analyzed, and the derived thermal conductance matches the universal quantum of heat conductance $G_{Q}$ in the quantum limit. Having a Hamiltonian scheme interpretable in thermodynamic terms, a Schrödinger-like wave equation is formulated whose wavefunction solutions contain the information about the entropy evolution. The results show exact agreement with 'classical' results for non abrupt changes. Finally, comparisons with a pure quantum mechanical treatment of the wave function in an expanding box confirm consistency in quasi-static regimes and reveal adiabaticity breakdown under far-from-equilibrium conditions.

Figures (8)

• Figure 1: $\Psi(g)$ for $\alpha=10$. Envelopes, in dashed lines, are normalized $\:^{+}_{-} f^{-1/2}$
• Figure 2: Derivative of $f^{0.5}\Psi$ and f(g)(dashed line)
• Figure 3: Probability density and $T^{-0.5}$ (dashed) for the evolution from $50 Te$ to $Te$ in normalized units.
• Figure 4: The surface is the solution $\Psi_{D}(x,t)$ found in Doescher, in red the same solution along the particle classical path $\Psi_{D}(x(g),t(g))$, in blue our solution $\Psi(x(g),t(g))$.
• Figure 5: Projections of the curves of Fig \ref{['fig:fig_Doesch_alf=50']} on the plane t=0. Only some of the blue curves were plotted for clarity.