An exactly solvable model for first- and second-order transitions

L.I. Klushin^a, A.M. Skvortsov^b, A.A. Gorbunov^c
^aInstitute of Macromolecular Compounds, Russian Academy of Sciences, Bolshoi prosp. 31, St. Petersburg, 199004, Russian Federation
^bSt. Petersburg Chemical-Pharmaceutical Academy, ul. prof. Popova 14, St. Petersburg, 197376, Russian Federation
^cState Research Institute for Highly Pure Biomaterials, ul. Pudozhskaya 7, St. Petersburg, 197110, Russian Federation

The possibility of an exact analytical description of first-order and second-order transitions is demonstrated using a specific microscopic model. Predictions using the exactly calculated partition function are compared with those based on the Landau and Yang-Lee approaches. The model employed is an adsorbed polymer chain with an arbitrary number of links and an external force applied to its end, for which the variation of the partition function with the adsorption interaction parameter and the magnitude of the applied force is calculated. In the thermodynamic limit, the system has one isotropic and two anisotropic, ordered phases, each of which is characterized by two order parameters and between which first-order and second-order transitions occur and a bicritical point exists. The Landau free energy is found exactly as a function of each order parameter separately and, near the bicritical point, as a function of both of them simultaneously. An exact analytical formula is found for the distribution of the complex zeros of the partition function in first-order and second-order phase transitions. Hypotheses concerning the way in which the free energy and the positions of the complex zeros scale with the number of particles N in the system are verified.

PACS: 05.70.Fh, 64.60.−i, 64.70.−p (all)
DOI: 10.1070/PU1998v041n07ABEH000417
URL: https://ufn.ru/en/articles/1998/7/b/
Web of Science

000075638400002
Citation: Klushin L I, Skvortsov A M, Gorbunov A A "An exactly solvable model for first- and second-order transitions" Phys. Usp. 41 639–649 (1998)

BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Клушин Л И, Скворцов А М, Горбунов А А «Точно решаемая модель, демонстрирующая фазовые переходы первого и второго рода» УФН 168 719–730 (1998); DOI: 10.3367/UFNr.0168.199807b.0719

References (30) Cited by (6) Similar articles (20) ↓

A.I. Olemskoi “Theory of stochastic systems with singular multiplicative noise” Phys. Usp. 41 269–301 (1998)
G.A. Martynov “The problem of phase transitions in statistical mechanics” Phys. Usp. 42 517–543 (1999)
M.A. Anisimov, E.E. Gorodetskii, V.M. Zaprudskii “Phase transitions with coupled order parameters” Sov. Phys. Usp. 24 57–75 (1981)
A.I. Olemskoi, I.V. Koplyk “The theory of spatiotemporal evolution of nonequilibrium thermodynamic systems” Phys. Usp. 38 1061–1097 (1995)
I.K. Kamilov, A.K. Murtazaev, Kh.K. Aliev “Monte Carlo studies of phase transitions and critical phenomena” Phys. Usp. 42 689–709 (1999)
L.I. Manevich, A.V. Savin et al “Solitons in nondegenerate bistable systems” Phys. Usp. 37 859–879 (1994)
S.M. Stishov “Quantum phase transitions” Phys. Usp. 47 789–795 (2004)
V.P. Skripov, A.V. Skripov “Spinodal decomposition (phase transitions via unstable states)” Sov. Phys. Usp. 22 389–410 (1979)
M.A. Mikulinskii “Effect of small perturbations on the behavior of thermodynamics quantities near a second-order phase transition point” Sov. Phys. Usp. 16 361–380 (1973)
B.M. Smirnov “Scaling method in atomic and molecular physics” Phys. Usp. 44 1229–1253 (2001)
V.I. Alkhimov “Excluded volume effect in statistics of self-avoiding walks” Phys. Usp. 37 527–561 (1994)
A.V. Khomenko, I.A. Lyashenko “Statistical theory of the boundary friction of atomically flat solid surfaces in the presence of a lubricant layer” Phys. Usp. 55 1008–1034 (2012)
A.V. Bushman, V.E. Fortov “Model equations of state” Sov. Phys. Usp. 26 465–496 (1983)
D.A. Kirzhnits, Yu.E. Lozovik, G.V. Shpatakovskaya “Statistical model of matter” Sov. Phys. Usp. 18 649–672 (1975)
S.V. Demishev “Spin-fluctuation transitions” Phys. Usp. 67 22–43 (2024)
N.E. Alekseevskii “New superconductors” Sov. Phys. Usp. 11 403–410 (1968)
A.Yu. Grosberg “Disordered polymers” Phys. Usp. 40 125–158 (1997)
I.M. Lifshits, A.Yu. Grosberg, A.R. Khokhlov “Volume interactions in the statistical physics of a polymer macromolecule” Sov. Phys. Usp. 22 123–142 (1979)
V.S. Dotsenko “Critical phenomena and quenched disorder” Phys. Usp. 38 457–496 (1995)
I.S. Lyubutin, A.G. Gavriliuk “Research on phase transformations in 3d-metal oxides at high and ultrahigh pressure: state of the art” Phys. Usp. 52 989–1017 (2009)

The list is formed automatically.