Thermodynamic Entropy as a Noether Invariant from Contact Geometry

Abstract

We use a formulation of Noether’s theorem for contact Hamiltonian systems to derive a relation between the thermodynamic entropy and the Noether invariant associated with time-translational symmetry. In the particular case of thermostatted systems at equilibrium, we show that the total entropy of the system plus the reservoir are conserved as a consequence thereof. Our results contribute to understanding thermodynamic entropy from a geometric point of view.

Why neural functionals suit statistical mechanics

“…[91] for an introductory presentation. Besides such deterministic applications, the Noether theorem is currently seeing an increased use in a variety of statistical mechanical settings [121][122][123][124][125][126][127][128].…”

Noether invariance theory for the equilibrium force structure of soft matter

“…This splitting discriminates between self and distinct contributions on the basis of the standard criterion for pairs of particle indices which are equal i = j (self) and different i ̸ = j (distinct). Hence splitting equation ( 53) leads respectively to the distinct sum rule (24) and, upon leaving away the common factor δ(r − r ′ ) from the self terms, to the self sum rule (25).…”

“…The potential forces then only arise from interparticle contributions and hence FU (r) = Fint (r) with the interparticle force density operator Fint (r) being given by equation (9). We address the distinct sum rule (24) and use the standard pair correlation function or 'radial distribution function' g(r). Here r = |r − r ′ | denotes the separation distance between the two particles.…”

“…We can now express equation (D5) succinctly and hence obtain the distinct part (24) of the curvature sum rule, which for completeness we supplement by re-writing the self part (25) (to be proven below):…”

“…Noether's theorem of invariant variations [16,17] was put in a statistical physics setting in a number of different ways [18][19][20][21][22][23][24]. The recent thermal Noether invariance theory [25][26][27][28][29][30][31] is based on the identification of the symmetry of the statistical many-body systems against specific shifting and rotation operations on phase space.…”