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SUMMARY:Microscopic Dynamical Entropy: Second Law from Hamiltonian Dynamic
 s - Mingnan Ding (Shanghai Jiao Tong University)
DTSTART:20250911T142500Z
DTEND:20250911T143000Z
UID:TALK233347@talks.cam.ac.uk
DESCRIPTION:Statistical mechanics seeks to derive macroscopic thermodynami
 cs from microscopic dynamics\, yet its central quantity&mdash\; entropy&md
 ash\; has long lacked a formulation that directly matches the thermodynami
 c second law. The second law of thermodynamics predicts increasing macrosc
 opic thermal entropy\, while the microscopic Gibbs entropy is conserved un
 der Hamiltonian dynamics. To resolve this discrepancy\, we introduce the M
 icroscopic Dynamical Entropy (MDE) for a system X coupled to a bath Y \, w
 ith the composite X + Y evolving under exact Hamiltonian dynamics. The MDE
  directly encodes the thermodynamic identity T∆S = ∆Q\, and coincides 
 with the Gibbs entropy when the bath distribution is taken to be uniform o
 n the energy shell. In this formulation\, the thermodynamic entropy increa
 se arises from discarding information into the bath&rsquo\;s degrees of fr
 eedom. The MDE preserves dynamics while establishing the second law and re
 solving the echo paradox. We demonstrate its consistency with both classic
 al and stochastic thermodynamics\, and show explicitly that finite baths i
 n the Zwanzig model already yield a monotonic increase of the MDE\, demons
 trating that irreversibility does not rely on the singular N &rarr\; &infi
 n\; limit.&nbsp\;
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