[Paper Review] Microscopic statistical basis of classical Thermodynamics of finite systems

D. H. E. Gross|arXiv (Cornell University)|May 10, 2005

Advanced Physical and Chemical Molecular Interactions17 references3 citations

TL;DR

This paper proposes a microcanonical statistical mechanics framework to describe phase transitions in finite, inhomogeneous systems—such as fragmenting nuclei and self-gravitating stellar systems—where traditional canonical ensembles fail. By avoiding the thermodynamic limit and redefining entropy via Boltzmann's microscopic states, it shows that negative heat capacity and irreversible internal entropy production are intrinsic, and that heat can flow from cold to hot during phase separation, invalidating classical assumptions of the second law in such contexts.

ABSTRACT

Heat can flow from cold to hot at any phase separation. Therefore Lynden-Bell's gravo-thermal catastrophe must be reconsidered. The original objects of Thermodynamics, the separation of phases at first order phase transitions, like boiling water in steam engines, are not described by a single canonical ensemble. Inter-phase fluctuations are not covered. The basic principles of statistical mechanics, especially of phase transitions have to be reconsidered without the use of the thermodynamic limit. Then thermo-statistics applies also to nuclei and large astronomical systems. A lot of similarity exists between the accessible phase space of fragmenting nuclei and inhomogeneous multi stellar systems.

Motivation & Objective

• To resolve the failure of canonical statistical mechanics in describing phase transitions in finite, inhomogeneous systems like fragmenting nuclei and self-gravitating systems.
• To reformulate thermodynamics without relying on the thermodynamic limit, which obscures phase separation and inter-phase fluctuations.
• To clarify the microscopic origin of entropy and the second law by distinguishing internal entropy production from external entropy exchange.
• To demonstrate that the second law remains valid even when heat flows from cold to hot during phase separation, as long as internal entropy production is considered.
• To establish a new foundation for statistical mechanics applicable to systems ranging from atomic clusters to galaxies, using microcanonical entropy and convexity of S(E)

Proposed method

• Uses the microcanonical ensemble to describe isolated systems with fixed energy, avoiding the canonical ensemble's assumption of thermal contact with a reservoir.
• Applies Boltzmann's microscopic definition of entropy: S = k_B ln W, where W is the number of accessible microstates for a given energy E.
• Analyzes the convexity of the entropy function S(E) at phase separations, which leads to negative heat capacity (dE/dT < 0) in the caloric curve T(E).
• Introduces the distinction between external entropy transfer (d_eS) and internal entropy production (d_iS), with d_iS ≥ 0 being the true indicator of irreversibility.
• Demonstrates that configurations with coexisting phases (e.g., liquid and gas in boiling water, or fragments in nuclear decay) cannot be described by a single canonical ensemble.
• Uses event-by-event statistical analysis of nuclear fragmentation data to identify phase-like peaks in the distribution, confirming the existence of inhomogeneous phases

Experimental results

Research questions

• RQ1 Can phase transitions in finite systems like fragmenting nuclei be described without the thermodynamic limit?
• RQ2 Why do standard canonical ensembles fail to describe phase separation and inter-phase fluctuations in finite systems?
• RQ3 How can the second law of thermodynamics be consistently formulated in microcanonical systems where heat may flow from cold to hot?
• RQ4 What is the role of internal entropy production (d_iS) in irreversible processes, and how does it differ from external entropy transfer?
• RQ5 Can negative heat capacity, observed in nuclear fragmentation and self-gravitating systems, be explained within a consistent statistical mechanical framework?

Key findings

• Phase separation in finite systems, such as boiling water or fragmenting nuclei, cannot be described by a single canonical ensemble due to inter-phase fluctuations.
• The second law remains valid even when heat flows from cold to hot during phase separation, as long as internal entropy production (d_iS) is considered.
• Negative heat capacity (C_V < 0) is a direct consequence of the convexity of the entropy function S(E) at phase separations, observed in nuclear fragmentation and self-gravitating systems.
• Microcanonical statistics reveals that irreversible processes like free expansion of an ideal gas increase entropy internally without heat exchange, confirming d_iS > 0.
• Event-by-event analysis of nuclear fragmentation data shows distinct peaks corresponding to inhomogeneous phases, confirming the existence of non-canonical, phase-separated states.
• The microcanonical framework explains bimodal mass distributions in nuclear fragmentation controlled by angular momentum, not just energy, extending beyond the liquid-gas transition