Analyzing fragmentation of simple fluids with percolation theory

Campi, Xavier; Krivine, H.; Sator, Nicolas; Plagnol, E.

doi:10.1007/s100530050028

Cited by 10 publications

(22 citation statements)

References 27 publications

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“…Performing molecular dynamics calculations with a Lennard-Jones fluid, Campi and his collaborators (Campi et al, 2001a;Sator, 2000) have found that the cluster size distributions obtained with these three definitions are very similar. To quote Hill (1955): "A cluster is [.…”

Section: A Local Criterion: Hill Clustersmentioning

confidence: 99%

“…It has to be noticed that the Hill's criterion can be used as well with Monte-Carlo calculations, although the velocities of the particles are not provided. Indeed, we checked by means of molecular dynamics simulations that mean cluster size distributions are not modified when we replace the velocity of each particle, provided by molecular dynamics simulations, with velocity components taken at random from a Gaussian distribution characterized by the temperature of the system (Sator, 2000). As far as mean cluster size distributions are concerned, position-velocity correlations are irrelevant and Monte-Carlo calculations can be used by allocating to each particle a velocity from a Gaussian distribution.…”

Section: A Local Criterion: Hill Clustersmentioning

confidence: 99%

“…With the help of canonical Monte Carlo and microcanonical molecular dynamics calculations (Allen and Tildesley, 1987), Campi and his collaborators (Sator, 2000;Campi et al, 2001a) have studied systems made up of a large number of particles (N = 11664) interacting through a Lennard-Jones potential given by Eq. (1).…”

Section: Percolation Lines In Simple Fluidsmentioning

confidence: 99%

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Clusters in simple fluids

Sator

2003

Physics Reports

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This article concerns the correspondence between thermodynamics and the morphology of simple fluids in terms of clusters. Definitions of clusters providing a geometric interpretation of the liquid-gas phase transition are reviewed with an eye to establishing their physical relevance. The author emphasizes their main features and basic hypotheses, and shows how these definitions lead to a recent approach based on self-bound clusters. Although theoretical, this tutorial review is also addressed to readers interested in experimental aspects of clustering in simple fluids.

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Section: A Local Criterion: Hill Clustersmentioning

confidence: 99%

Section: A Local Criterion: Hill Clustersmentioning

confidence: 99%

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Clusters in simple fluids

Sator

2003

Physics Reports

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“…Percolation models [255,256,257,258], the Fisher Droplet Model [221,226,259,260,261] and more recently the theory of universal fluctuations [207,225,262,263,264] have been employed to describe fragment size distributions and fluctuations and to tentatively derive information on the critical region of the liquid-gas transition: in finite systems the critical point becomes a critical region. The determinations of a scaling function and of a consistent set of critical exponents in different multifragmentation data also tend to support this hypothesis [97,141,265,266].…”

Section: Scaling and Fluctuations For Fragment Sizesmentioning

confidence: 99%

Nuclear multifragmentation and phase transition for hot nuclei

Borderie

Rivet

2008

Progress in Particle and Nuclear Physics

202

174

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That review article is focused on the tremendous progresses realized during the last fifteen years in the understanding of multifragmentation and its relationship to the liquid-gas phase diagram of nuclei and nuclear matter. The explosion of the whole nucleus, early predicted by Niels Bohr [1], is a very complex and rich subject which continues to fascinate nuclear physicists as well as theoreticians who extend the thermodynamics of phase transitions to finite systems.

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“…Fragmentation is a widely studied phenomena [1] with applications ranging from conventional fracture of solids [2] and collision induced fragmentation in atomic nuclei/aggregates [3] to seemingly unrelated fields such as disordered systems [4] and geology [5]. In this paper we consider a problem where an object of initial size (or length) x is first broken into m random pieces of sizes x i = r i x with m i=1 r i = 1 provided the initial size x > x 0 where x 0 is a fixed 'atomic' threshold.…”

mentioning

confidence: 99%

Phase transition in a random fragmentation problem with applications to computer science

Dean

Majumdar

2002

J. Phys. A: Math. Gen.

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We study a fragmentation problem where an initial object of size x is broken into m random pieces provided x > x0 where x0 is an atomic cut-off. Subsequently the fragmentation process continues for each of those daughter pieces whose sizes are bigger than x0. The process stops when all the fragments have sizes smaller than x0. We show that the fluctuation of the total number of splitting events, characterized by the variance, generically undergoes a nontrivial phase transition as one tunes the branching number m through a critical value m = mc. For m < mc, the fluctuations are Gaussian where as for m > mc they are anomalously large and non-Gaussian. We apply this general result to analyze two different search algorithms in computer science. [5]. In this paper we consider a problem where an object of initial size (or length) x is first broken into m random pieces of sizes x i = r i x with m i=1 r i = 1 provided the initial size x > x 0 where x 0 is a fixed 'atomic' threshold. At the next stage, each of those m pieces with sizes bigger than x 0 is further broken into m random pieces and so on. Clearly the process stops after a finite number of fragmentation or splitting events when the sizes of all the pieces become less than x 0 . This problem and its close cousins have already appeared in numerous contexts including the energy cascades in turbulence [6], rupture processes in earthquakes [7], stock market crashes [8], binary search algorithms [9-11], stochastic fragmentation [12] and DNA segmentation algorithms [13]. It therefore comes as somewhat of a surprise that there is a nontrivial phase transition in this problem as one tunes the branching number m through a critical value m = m c .In this Letter we study analytically the statistics of the total number of fragmentation events n(x) up till the end of the process as a function of the initial size x. We show that, while the average number of events µ(x) always grows linearly with x for large x, the asymptotic behavior of the variance ν(x), characterizing the fluctuations, undergoes a phase transition at a critical value m = m c ,The exponent θ is nontrivial and increases monotonically with m for m ≥ m c starting at θ(m = m c ) = 1/2 and the amplitude of the leading x 2θ term has log-periodic oscillations for m ≥ m c . This signals unusually large fluctuations in n(x) for m > m c . The full distribution of n(x) also changes from being Gaussian for m < m c to non-Gaussian for m > m c . This phase transition is rather generic for any fragmentation problem with an 'atomic' threshold. However the critical value m c and the exponent θ are nonuniversal and depend on the distribution function of the random fractions r i 's. In this Letter we establish this generic phase transition and then calculate explicitly m c and θ for two special cases with direct applications in computer science.In this fragmentation problem with a fixed lower cut-off x 0 , one first breaks the initial piece of length x provided x > x 0 into m pieces of sizes x i = r i x. The sizes of each of...

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Analyzing fragmentation of simple fluids with percolation theory

Cited by 10 publications

References 27 publications

Clusters in simple fluids

Clusters in simple fluids

Nuclear multifragmentation and phase transition for hot nuclei

Phase transition in a random fragmentation problem with applications to computer science

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