About

Paul Beale is a Professor of Physics at the University of Colorado Boulder, where he has held the position since 1997. His general field of research is the thermodynamics and statistical mechanics of condensed matter systems. His work includes theoretical studies related to critical phenomena and phase transitions, ferroelectrics, Landau theories, hysteresis in spin models, magnetic materials, failure modes in random materials, theories of melting, effects of noise on nonlinear dynamical systems, phenomenological finite-size scaling, Monte Carlo methods, commensurate-incommensurate transitions, renormalization-group methods, and structural phase transitions. His recent contributions include a calculation of the exact distribution of energies in the two-dimensional Ising model, utilizing Kauffman's generalization of Onsager's solution to compute the exact form of the partition function in a low-temperature series expansion. This provides an exact distribution that verifies the accuracy and convergence of Monte Carlo simulations. Additionally, he has developed a new class of pseudorandom number generators based on Pohlig-Hellman exponentiation ciphers, which generate uniform pseudorandom streams with advantages such as scalability, simplicity, and passing extensive correlation tests. His work also encompasses various computational and theoretical methods, including the development of Mathematica routines for the exact partition function of the 2D Ising model and research on the breakdown of two-phase random resistor networks, dielectric breakdown, and the properties of ferroelectric switching.

Research topics

• Physics
• Mathematics
• Statistical physics
• Mathematical analysis
• Geometry
• Mathematical physics
• Quantum mechanics
• Condensed matter physics

Selected publications

• Replication Data for: Traceable random numbers from a non-local quantum advantage

  CORA.Repositori de Dades de Recerca · 2026-03-13

  datasetOpen access

  The unpredictability of random numbers is fundamental to both digital security1,2 and applications that fairly distribute resources3,4. However, existing random number generators have limitations—the generation processes cannot be fully traced, audited and certified to be unpredictable. The algorithmic steps used in pseudorandom number generators5 are auditable, but they cannot guarantee that their outputs were a priori unpredictable given knowledge of the initial seed. Device-independent quantum random number generators6,7,8,9 can ensure that the source of randomness was unknown beforehand, but the steps used to extract the randomness are vulnerable to tampering. Here we demonstrate a fully traceable random number generation protocol based on device-independent techniques. Our protocol extracts randomness from unpredictable non-local quantum correlations, and uses distributed intertwined hash chains to cryptographically trace and verify the extraction process. This protocol forms the basis for a public traceable and certifiable quantum randomness beacon that we have launched10. Over the first 40 days of operation, we completed the protocol 7,434 out of 7,454 attempts—a success rate of 99.7%. Each time the protocol succeeded, the beacon emitted a pulse of 512 bits of traceable randomness. The bits are certified to be uniform with error multiplied by actual success probability bounded by 2−64. The generation of certifiable and traceable randomness represents a public service that operates with an entanglement-derived advantage over comparable classical approaches.

  PublisherDOI

• Traceable random numbers from a non-local quantum advantage

  Nature · 2025-06-11 · 6 citations

  article
  PublisherDOI

• Traceable random numbers from a nonlocal quantum advantage

  arXiv (Cornell University) · 2024-11-08

  preprintOpen access

  The unpredictability of random numbers is fundamental to both digital security and applications that fairly distribute resources. However, existing random number generators have limitations-the generation processes cannot be fully traced, audited, and certified to be unpredictable. The algorithmic steps used in pseudorandom number generators are auditable, but they cannot guarantee that their outputs were a priori unpredictable given knowledge of the initial seed. Device-independent quantum random number generators can ensure that the source of randomness was unknown beforehand, but the steps used to extract the randomness are vulnerable to tampering. Here, for the first time, we demonstrate a fully traceable random number generation protocol based on device-independent techniques. Our protocol extracts randomness from unpredictable non-local quantum correlations, and uses distributed intertwined hash chains to cryptographically trace and verify the extraction process. This protocol is at the heart of a public traceable and certifiable quantum randomness beacon that we have launched. Over the first 40 days of operation, we completed the protocol 7434 out of 7454 attempts -- a success rate of 99.7%. Each time the protocol succeeded, the beacon emitted a pulse of 512 bits of traceable randomness. The bits are certified to be uniform with error times actual success probability bounded by $2^{-64}$. The generation of certifiable and traceable randomness represents one of the first public services that operates with an entanglement-derived advantage over comparable classical approaches.

  PublisherOA PDFDOI

• Ideal Bose systems

  Elsevier eBooks · 2021-02-21

  book-chapterSenior author
  PublisherDOI

• Statistical mechanics of interacting systems: the method of quantized fields

  Elsevier eBooks · 2021-02-21

  book-chapterSenior author
  PublisherDOI

• Statistical mechanics of interacting systems: the method of cluster expansions

  Elsevier eBooks · 2021-02-21

  book-chapterSenior author
  PublisherDOI

• Statistical Mechanics Third Edition

  2021 · 22 citations

  1st authorCorresponding
  • Statistical physics
  • Physics
  Publisher

• The canonical ensemble

  Elsevier eBooks · 2021-02-21 · 1 citations

  book-chapterSenior author
  PublisherDOI

• The statistical basis of thermodynamics

  Elsevier eBooks · 2021-02-21 · 1 citations

  book-chapterSenior author
  PublisherDOI

• Phase transitions: the renormalization group approach

  Elsevier eBooks · 2021-02-21 · 1 citations

  book-chapterSenior author
  PublisherDOI

Frequent coauthors

• R. K. Pathria

  University of California, San Diego

  22 shared

• Charles T. Rogers

  University of Colorado Boulder

  12 shared

• R.K. Pathria

  12 shared

• Noel A. Clark

  University of Colorado Boulder

  11 shared

• Shawn C. Gay

  9 shared

• Jiřı́ Kaleta

  Czech Academy of Sciences, Institute of Organic Chemistry and Biochemistry

  8 shared

• Steven J. Pollock

  University of Colorado Boulder

  8 shared

• Josef Michl

  University of Colorado Boulder

  8 shared