Professor Oliver Penrose, F.R.S., F.R.S.E.

Oliver Penrose was a Professor of Mathematics at Heriot-Watt University from 1986 until his retirement in 1994. Before that he spent 17 years at the Open University. He is still active in research and now holds the title Professor Emeritus.

His principal research interest is in Statistical Mechanics, both equilibrium and non-equilibrium, and in related topics such as the kinetics of phase transitions in metals and the physical chemistry of surfactants; he makes occasional forays into more metaphysical areas such as the direction of time and the interpretation of quantum mechanics.

His book `Foundations of statistical mechanics' (Pergamon 1970) has been re-issued as a Dover reprint.

His recreations include music and chess

His personal (non-scientific) web site is

Lecture notes

The Becker-Doering equations for the kinetics of phase transitions ( Strathclyde University, 1995) (pdf file).

Statistical mechanics for mathematicians (Heriot-Watt University, January-February 2003)

  • Lecture 1 (text file).
  • Lecture 2 (text file).
  • Lecture 3 (text file).
  • Lecture 4 (text file).
  • Lecture 5 (pdf file).
  • Lecture 6 (pdf file).

    Statistical mechanics of nonlinear elasticity in a system of hard disks (ICMS conference "Geometric Analysis, Elasticity and PDE" in honour of J M Ball's 65th birthday, Edinburgh, June 2008.) Click here to see the transparencies I made for this lecture: (pdf file )

    Mathematical modelling, finance and the recession (BMC/BAMC, Edinburgh, 6-9 April 2010) Click here to see the electronic slides I made for this presentation: (pdf file). Click here to see the slides used for my longer talk with the same title at Heriot-Watt University 26 May 2010 (pdf file). Click here to see a version of this talk in the form of a text with illustrations (pdf file).

    Positive thinking about negative temperatures (conference "Analytical results in statistical physics" in memory of Bernard Jancovici, Paris November 5-6 2015). Click here to see the slides used for this talk (pdf file).

    The paradox of irreversibility, talk at University of Bath, January 2017. To see the slides used for the talk click here (pdf file).

    An application of centre manifold theory to economics, talk at the conference in memory of Jack Carr held at Heriot-Watt University in May 2018. To see the slides used in this talk click here (pdf file).

    Projects in progress or abandoned

    Statistical mechanics of nonlinear elasticity (see reference 19 below and lecture notes above)

    Theory of curvature and grooving effects in diffusion-induced grain boundary motion, (with J W Cahn). I regret to say that this project is unlikely to be completed, since John Cahn died in March 2016

    Extension of the theory described in my book `Foundations of statistical mechanics' (Dover reprint, 2005) to the case where the sequence of successive obervational states is not assumed to be Markovian

    Off-diagonal long-range order in its relation to the thermodynamics and electrodynamics of superconductivity

    Time evolution of a weakly interacting Bose system, with or without condensate (Lecture at Warwick University workshop on 'Large many-body systems', 23 August 2004) (pdf file)

    A system of kinetic equations for phase transitions in the presence of mushy zones

    "Statistical mechanics" of two-dimensional turbulence ( pdf file)

    Theory of one-component micelles (with D.G. Hall and J. Wattis), unlikely to be completed.

    Time's arrow and Occam's razor (incomplete draft, 2015 or earlier) (pdf file)

    Ancient Publications

    1. Bose-Einstein Condensation & Phonons in Helium II. Paper presented at the Fifth International Conference on Low-temperature Physics and Chemistry, Madison, Wisconsin, August 26-31, 1957, published in "Low-Temperature Physics and Chemistry", ed. J R Dillinger, University of Wisconsin Press, Madison, 1958 (pdf file)

      Recent Publications

      1. Quantum mechanics and real events, O. Penrose, pp 257-264 of Quantum Chaos - Quantum Measurement edited by P Cvitanovic et al. (Kluwer, 1992) ( pdf file)
      2. Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel. M Kreer and O.Penrose, J. Stat. Phys. 75, 389-407 (1994). (pdf file)
      3. Interfacial dynamics for thermodynamically consistent phase-field models with nonconserved order parameter. P Fife and O Penrose, Electronic J. Diff. Eqns. Vol. 1995 (1995) no.16 pp 1-49. (US sites: Abstract, Postscript File , TeX code, etc.. European sites: Abstract, Postscript File, TeX code, etc.). ( pdf file)
      4. Motion resulting from diffusion within an interface: J. W. Cahn, P. C. Fife and O. Penrose. pp 37-46 of Curvature Flows and Related Topics (Proceedings of a conference held in Levico in 1994) ed. A. Damlamian, J. Spruck and A. Visintin (Gakkotosho, Tokyo, 1995)
      5. Metastable decay rates, asymptotic expansions, and analytic continuation of thermodynamic functions: O. Penrose, J.Stat.Phys 78, 267-283 (1995). (postscript file).
      6. Ising model for phase separation in alloys with elastic interaction. I Theory: P. Fratzl and O. Penrose, Acta Metall. Mater. 43, 2921-2930 (1995). (postscript file).
      7. Ising model for phase separation in alloys with anisotropic elastic interaction.II A computer experiment: P. Fratzl and O. Penrose, Acta Materiala 44, 3227-3229 (1996) (postscript file).
      8. The "game of everything": O. Penrose, Markov Processes Relat. Fields 2, 167-182 (1996) (PDF file).
      9. Kinetics of spinodal decomposition in the Ising model with vacancy dynamics - Reply: P. Fratzl and O. Penrose, Physical Review B 53, 2890-2891 (1996).
      10. The Wiener programme in statistical physics: is it feasible in the light of recent developments? Proceedings of the Norbert Wiener Centenary Congress, 1994 (eds V Mandrekar and P R Masani) AMS Proc. Symp. Appl. Math. 52, 37-52 (1997).
      11. Dissolution of precipitates heated above the solubility line: A Monte Carlo simulation: I Zizak, P. Fratzl and O. Penrose, Phys Rev B 55, 12121-12127 (1997).
      12. Competing mechanisms for precipitate coarsening in phase separation with vacancy dynamics: P. Fratzl and O. Penrose, Phys. Rev. B 55, R6101-R6104 (1997).
      13. The Becker-Doering equations at large times and their connection with the LSW theory of coarsening: O. Penrose, J. Stat. Phys. 89, 305-320 (1997)
      14. A phase-field model for diffusion-induced grain boundary motion: J. W. Cahn, P. C. Fife and O. Penrose, Acta Mater. 45, 4397-4413 (1997).
      15. Asymptotic behaviour of solutions to a simplified Lifshitz-Slyozov equation. J. Carr and O. Penrose, Physica D 124, 166-176 (1998).
      16. Modeling of phase separation in alloys with coherent elastic misfit: Peter Fratzl, Oliver Penrose and Joel L. Lebowitz, J. Stat. Phys. 95, 1429-1503 (1999).
      17. Coarsening in the Ising model with vacancy dynamics: P.Fratzl, O.Penrose, R.Weinkamer and I.Zizak, Physica A 279, 100-109 (2000).
      18. Close to close packing: O. Penrose and G. Stell, J. Stat. Phys. 100, 89-95 (2000) (pdf file).
      19. The direction of time: pp 61-82 of the book "Chance in physics: foundations and perspectives" (proceedings of a conference held in Naples, 1999) edited by J. Bricmont and others, Springer lecture notes in physics no. 574 (2001) (pdf file).
      20. Statistical mechanics of nonlinear elasticity: Markov Processes and Related Fields 8, 351-364 (2002) (pdf file); correction: ibid. 12, 169 (2006) (pdf file).
      21. A mathematical model for diffusion-induced grain boundary motion, by O Penrose and J W Cahn, pp 237-254 of "Free Boundary Problems: Theory and Applications" (ed. P, Colli, C. Verdi and A. Visintin), International Series of Numerical Mathematics Vol. 147, Birkhauser, Basel 2003 (pdf file).
      22. Using kinetic Monte Carlo simulations to study phase separation in alloys: R. Weinkamer, P. Fratzl, H.S. Gupta, O. Penrose, and J.L. Lebowitz, Phase Transitions 77, 433-456 (2004). (pdf file).
      23. On the elastic driving mechanism in diffusion-induced grain boundary motion: O. Penrose, Acta Materialia 52, 3901-3910 (2004) (pdf file).
      24. A family of balance relations for the two-dimensional Navier-Stokes equations with random forcing: S. Kuksin and O. Penrose, J. Stat. Phys. 118, 437-449 (2005) ( pdf file)
      25. An asymmetric world: O. Penrose, Nature 438, 919 (2005) ( pdf file)
      26. Reversibility and irreversibility: O. Penrose, Oberwolfach Reports 3, 2682-2685 (European Mathematical Society 2006). ( pdf file )
      27. A beautiful method of analysis: O. Penrose, pp 443-451 of "Fifty years of human genetics: A Festschrift and liber amicorum to celebrate the life and work of George Robert Fraser", edited by Oliver Mayo and Carolyn Leach, (ISBN 9781862547537, Wakefield Press, Adelaide, 2007)
      28. Kelvin and statistical mechanics : chapter 14 of "Kelvin: life, labours and legacy" edited by Raymond Flood, Mark McCartney and Andrew Whitaker, (ISBN-13: 978-0199231256, Oxford University Press, Oxford 2008)
      29. Nucleation and droplet growth as a stochastic process : chapter 12 (pp 265-277) of "Analysis and Stochastics of Growth Processes", edited by P. Morters, R. Moser, M. Penrose, H. Schwetlick and J. Zimmer, (ISBN 978-0-19-923925, Oxford University Press, Oxford 2008.) (pdf file)
      30. How to beat the recession: public enterprise, not fiscal stimulus: Compass Thinkpiece no. 51 (May 2009). ( pdf file)
      31. Probabilistic characteristics of random damage events and their quantification in acrylic bone cement : G. Qi, S. F. Wayne, O. Penrose, G. Lewis, J. Hochstein and K. Mann, Journal of Materials Science: Materials in Medicine (2010) 21:2915-2922 DOI 10.1007/s10856-010-4155-9 (pdf file)
      32. Comparing the efficiencies of stochastic isothermal molecular dynamics methods : B. Leimkuhler, E. Noorizadeh and O. Penrose, J. Stat. Phys. 143 (2011), 921-942. DOI: 10.1007/s10955-011-0210-2 (pdf file)
      33. Entropy and irreversibility in dynamical systems : O.Penrose, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences (2013) 2013371 20120349;DOI: 10.1098/rsta.2012.0349. (pdf file )
      34. Physics of negative absolute temperatures : E.Abraham and O.Penrose, Physical Review E 95 (Jan. 2017), 012125-1 to -8. DOI: 10.1103/PhysRevE.95.012125 (pdf file )
      35. On the mathematical modelling of cellular (discontinuous) precipitation : O. Penrose and the late J. W. Cahn : Discrete and Continouous Dynamical Systems, series A (DCDS-A) 37 (Feb. 2017) 963 - 982 ; doi:10.3934/dcds.2017040 (pdf file )
      36. Microscopic Irreversibility: Looking for a Microscopic Description of Time Asymmetry : O.Penrose, Journal of Statistical Physics (published on-line 4 May 2020)

      For a full list of publications up to December 2005 please click here : ( pdf file).

      Oliver Penrose / Heriot-Watt University / / tel. +44 131 451 3225 /
      fax +44 131 451 3249