Symmetry and Symmetry Breaking in Statistical Systems

Dear Colleagues,

Background: How things change in nature is controlled by two separate factors: the initial conditions, which may be unpredictable, and the laws that synthesize the regularity, but may be hard to discover because of irregularities produced by uncontrollable external factors. It is in the discovery of laws that symmetry principles play a crucial role. As external factors are central to statistical systems, symmetry principles also play an important role. Gibbs had already realized the importance of permutation symmetry in clarifying the paradox named after him. The symmetry principles of physical laws are modified by the statistics of large systems and determine the properties of the latter. While the role of symmetry principles in equilibrium statistical mechanics has been understood for quite some time, their role in non-equilibrium statistical physics is just as important, as is evidenced by Onsager’s reciprocity relations and by the Curie symmetry principle; the latter immediately leads to the idea of symmetry breaking, which permeates the entire field of statistical physics. Accordingly, while the basic laws governing a system have certain symmetries, some or all of them need not be present in the observed behavior. Despite this, the missing symmetries have consequences for the system, such as the emergence of Goldstone bosons (phonons, magnons, pions, etc.) and control phase transitions. In addition, new symmetries, such as scale invariance, can emerge thermodynamically near a critical point.

Scope: The symmetry principles (global or local as in gauge symmetries) should, in general, control the static and dynamic properties of all statistical systems including, but not limited to, equilibrium phases, phase transitions (classical and quantum), non-equilibrium states, transport, distinction between heat and work, fluctuations close to or far from equilibrium, quantum systems (closed or open), nature of coarse graining, topological phases, etc.

Recent Trends: Recently, symmetry operations have been used in identifying entangled states to better understand quantum phase transitions. In addition, considerable attention has been paid to understanding the consequences of symmetry on entanglement between a system and the surrounding medium as part of developing the principles of quantum thermodynamics. Another recent field of activity relates to various fluctuation theorems, which strongly constrain the nature of far from equilibrium fluctuations. As the fluctuations are governed by the Hamiltonian dynamics, which operate at the microscopic level, the symmetry of the Hamiltonian must also have ramifications for the fluctuations. Stochastic thermodynamics exploiting microstate evolution is another active field being pursued recently. There has been some effort to microscopically explain heat and work in terms of microstates. The application of AdS/CFT to conformal field theories has also received attention in string theory and model systems near a critical point.

Aim: My hope is to have contributions on various symmetries, those that are part of laws and those that are generated thermodynamically (also called dynamically), such as near a second order phase transition, in classical and quantum statistical physics applied to condensed matter, particle physics, black hole thermodynamics, etc., to provide a comprehensive perspective of the usefulness of symmetry principles, including recent cutting-edge trends, some of which are noted above.

Prof. Dr. Purushottam D. Gujrati
Guest Editor

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• space group symmetries,
• time-reversal symmetry,
• permutation,
• liquid crystal symmetries,
• scale invariance,
• conformal invariance,
• gauge symmetry,
• supersymmetry,
• chiral symmetry,
• symmetry breaking and restoration,
• material frame indifference and covariance transformation;
• heat and work,
• adiabatic invariance,
• temperature in relativistic thermodynamics,
• entanglement,
• fluctuation theorems,
• stochastic thermodynamics,
• quantum thermodynamics,
• black hole thermodynamics;
• holographic principle;
• AdS/CFT