Dominance of rare events in some problems in statistical physics

Saha, Arnab; Chakravarty, Sagar; Bhattacharjee, Jayanta K.

doi:10.1007/s12043-008-0175-0

“…Clearly the histograms are not symmetric about the marked r m , e.g., are not Gaussians. Log-normal curves 47 for τ 4 > τ > τ 2 , shown as solid lines, are closer matches. Below τ 2 , the histograms narrow sharply, as shown in the delta-function-like peak on the right.…”

Section: B Distributionsmentioning

confidence: 65%

“…47 Kolmogorov has also argued that the size distribution of rock fragments produced by sequential and random splittings should be log-normal. The rate distribution can be similarly understood, heuristically.…”

Section: B Distributionsmentioning

confidence: 99%

“…It is the logarithm of the rate, ln r m ∼ ln = i ln p i , that is the sum of a large number of independent, identically distributed stochastic variables, and by the central limit theorem it is this logarithm that will have a Gaussian distribution. 47 It is thus reasonable for conversion rates through increasingly rare search pathways to have the log-normal distribution of Eq. (5.3b) and Fig.…”

Section: B Distributionsmentioning

confidence: 99%

See 1 more Smart Citation

Monte Carlo simulations of strain pseudospins: Athermal martensites, incubation times, and entropy barriers

Shankaraiah

¹

,

Murthy

²

,

Lookman

³

et al. 2011

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We study martensitic transition kinetics through temperature-quench Monte Carlo simulations for a square/rectangle ferroelastic transition, described by a Hamiltonian of three-state pseudospins S, without extrinsic disorder. Here S = 0 for high-temperature austenite, and S = ±1 for the two martensite variants. The temperaturedependent pseudospin Hamiltonian comes from the total scaled free energy functional, evaluated at the three minima of Landau polynomials in order-parameter strains. It includes power-law anisotropic interactions from the St. Venant compatibility constraint, which orient the elastic domain walls in a symmetry-breaking diagonal direction. We find that temperature-time-transformation (TTT) plots for domain-wall phase evolution have phase crossover temperatures, which are understood through an effective-droplet energy parametrization. For temperature cycling through the phases, there are hysteretic peaks in physical quantities. For temperature quenches, a "vapor" of martensitic droplets converts at a time t m (T ) to a vibrating "liquid" of bidiagonal domain walls, which then orient at a time t C (T ) to a static "crystal" of single-diagonal martensitic twins, which can have bound residual austenite. Focusing on the conversion time t m , we find a material-parameter phase diagram, which has regions of nonactivated "athermal" and activated "isothermal" martensites. In an athermal, nonactivated regime, there are explosive austenite-martensite conversions at temperatures below a residual-austenite spinodal in the TTT diagram, while above it, there are conversion tails, as in experiment. We find t m (T ) has a quasi-universal Vogel-Fulcher divergence at transition, with a log-normal conversion-rate distribution. The incubation times t m , t C are attributed to entropy barriers, with signatures of flat energies, during pathway searches for finite-scale transition textures, which are explicitly identified through textural and internal-stress snapshots. In a glasslike energy landscape picture, these entropic pathways to many locked-twin states dominate those to a single-variant martensite state, of almost the same energy density. Other transitions in 2D and 3D can be similarly studied. More generally, the models could be used to explore conceptual issues of how systems equilibrate after a deep quench.

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“…The rapid and slow incubation-delay conversions are found to arise from the presence of large and small size of the golf-course edges. The VogelFulcher conversion-delays that are insensitive to Hamiltonian energy scales are found to have Log-normal distributions that are signatures of rare events 33 . The number of successful conversions, that are also insensitive to energy scales, vanishes where the entropy barriers diverges 12,17,20 .…”

Section: Energy Landscape Conceptsmentioning

confidence: 92%

Golf-course and funnel energy landscapes: Protein folding concepts in martensites

Shankaraiah

¹

2017

Phys. Rev. E

0

1

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We use protein folding energy landscape concepts such as golf-course and funnel to study reequilibration in athermal martensite parameter regime of triangle-to-centered rectangle, squareto-oblique, and triangle-to-oblique transitions under systematic temperature-quench Monte Carlo simulations. On quenching below a transition temperature, the seeded high-symmetry parent-phase austenite that converts to the low-symmetry product-phase martensite, through autocatalytic twinning or elastic photocopying, has both rapid conversions and incubation-delays in the temperaturetime-transformation phase diagram. We find the rapid (incubation-delays) conversions at low (high) temperatures arises from the presence of large (small) size of golf-course edge that has funnel inside for negative energy states. In the incubating state, the strain structure factor enters into the Brillouin zone golf-course through searches for finite transitional pathways which closes off at the transition temperature with Vogel-Fulcher divergences that are insensitive to Hamiltonian energy scales and log-normal distributions, as signatures of dominant entropy barriers. The crossing of the entropy barrier is identified through energy occupancy distributions, Monte Carlo acceptance fractions, heat emission and internal work. The above ideas had previously been presented for the scalar order parameter case. Here we show similar results are also obtained for vector order parameters.

show abstract

“…The rapid and slow incubation-delay conversions are found to arise from the presence of large and small size of the golf-course edges. The Vogel-Fulcher conversion-delays that are insensitive to Hamiltonian energy scales are found to have Log-normal distributions that are signatures of rare events 33 . The number of successful conversions, that are also insensitive to energy scales, vanishes where the entropy barriers diverges 12,17,20 .…”

Section: Introductionmentioning

confidence: 91%

Golf-course and funnel energy landscapes: Protein folding concepts in martensites

Shankaraiah

2017

Preprint

0

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We use protein folding energy landscape concepts such as golf-course and funnel to study reequilibration in athermal martensite parameter regime of triangle-to-centered rectangle, squareto-oblique, and triangle-to-oblique transitions under systematic temperature-quench Monte Carlo simulations. On quenching below a transition temperature, the seeded high-symmetry parent-phase austenite that converts to the low-symmetry product-phase martensite, through autocatalytic twinning or elastic photocopying, has both rapid conversions and incubation-delays in the temperaturetime-transformation phase diagram. We find the rapid (incubation-delays) conversions at low (high) temperatures arises from the presence of large (small) size of golf-course edge that has funnel inside for negative energy states. In the incubating state, the strain structure factor enters into the Brillouin zone golf-course through searches for finite transitional pathways which closes off at the transition temperature with Vogel-Fulcher divergences that are insensitive to Hamiltonian energy scales and log-normal distributions, as signatures of dominant entropy barriers. The crossing of the entropy barrier is identified through energy occupancy distributions, Monte Carlo acceptance fractions, heat emission and internal work. The above ideas had previously been presented for the scalar order parameter case. Here we show similar results are also obtained for vector order parameters.

show abstract

Dominance of rare events in some problems in statistical physics

Abstract: We show how the theory of large deviations in the coin toss experiment can give some insight into nonequilibrium fluctuation theorems and intermittency in turbulence.

Cited by 5 publications

References 22 publications

Monte Carlo simulations of strain pseudospins: Athermal martensites, incubation times, and entropy barriers

Monte Carlo simulations of strain pseudospins: Athermal martensites, incubation times, and entropy barriers

Golf-course and funnel energy landscapes: Protein folding concepts in martensites

Golf-course and funnel energy landscapes: Protein folding concepts in martensites

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