Covariant statistical mechanics and the stress-energy tensor

Area of Science:

• Relativistic statistical mechanics
• Thermodynamics
• Quantum field theory

Background:

• Equilibrium statistical mechanics in special relativity is crucial for understanding particle behavior.
• The covariant formalism needs extension to include non-vanishing spin tensors.
• Relativistic stress-energy tensor is fundamental in general relativity and thermodynamics.

Purpose of the Study:

• To extend the covariant formalism of equilibrium statistical mechanics to include a non-vanishing spin tensor.
• To derive the relativistic stress-energy tensor at thermodynamical equilibrium.
• To establish a connection between the stress-energy tensor and entropy current.

Main Methods:

• Recapitulation of the covariant formalism of equilibrium statistical mechanics in special relativity.
• Extension of the formalism to include a non-vanishing spin tensor.
• Functional derivative of the partition function with respect to the inverse temperature four-vector (β).

Main Results:

• The relativistic stress-energy tensor at thermodynamical equilibrium is obtained via a functional derivative.
• A formula T(μν) = -∂Φ(μ)/∂β(ν) is derived, relating stress-energy tensor to the relativistic thermodynamic potential current.
• This establishes a direct link between the stress-energy tensor and the entropy current at equilibrium.

Conclusions:

• A novel method for calculating the relativistic stress-energy tensor at equilibrium has been established.
• The derived formula provides a new perspective on the relationship between energy-momentum and entropy.
• The formalism may be extendable to nonequilibrium hydrodynamics, offering potential for future research.