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A note on Lee-Yang zeros in the negative half-plane.

Joel L Lebowitz1, Jasen A Scaramazza

  • 1Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA. Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854, USA.

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|August 23, 2016
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Summary

We establish lower bounds for inverse compressibility in systems with Lee-Yang zeros in the complex fugacity plane. This research explores the virial expansion for such systems, finding no direct link between positive virial coefficients and negative Lee-Yang zeros.

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Area of Science:

  • Statistical Mechanics
  • Phase Transitions
  • Lattice Models

Background:

  • Lee-Yang zeros of the grand-canonical partition function are crucial for understanding phase transitions and critical phenomena in statistical systems.
  • The location of these zeros in the complex fugacity plane provides insights into the system's thermodynamic properties and stability.
  • The monomer-dimer system on a lattice is a well-studied model exhibiting interesting phase behavior.

Purpose of the Study:

  • To derive lower bounds on the inverse compressibility for systems whose Lee-Yang zeros are located in the left half of the complex fugacity plane.
  • To investigate the virial expansion of pressure in terms of density for systems with Lee-Yang zeros on the negative real axis, such as the monomer-dimer system.
  • To explore the relationship between the positivity of virial coefficients and the location of Lee-Yang zeros.

Main Methods:

  • Analytical derivation of lower bounds for inverse compressibility based on the properties of Lee-Yang zeros.
  • Analysis of the virial expansion of pressure for specific lattice models.
  • Explicit calculation of the partition function for the monomer-dimer system on two rows.

Main Results:

  • Established lower bounds on the inverse compressibility for systems with Lee-Yang zeros in the left complex fugacity plane, including those on the negative real axis.
  • Demonstrated that there is no direct correlation between the positivity of virial coefficients and the location of Lee-Yang zeros in the negative real axis.
  • Provided examples illustrating cases where Lee-Yang zeros are negative, virial coefficients are positive, or both conditions are met.

Conclusions:

  • The study provides theoretical bounds on a key thermodynamic quantity, inverse compressibility, linked to Lee-Yang zero locations.
  • The research clarifies the complex relationship between Lee-Yang zeros and virial coefficients, showing they are not directly coupled.
  • An explicit calculation for the monomer-dimer system suggests a finite number of negative virial coefficients, offering specific insights into lattice models.