Rigorous Inequalities Between Length and Time Scales in Glassy Systems

Montanari, Andrea; Semerjian, Guilhem

doi:10.1007/s10955-006-9175-y

Cited by 225 publications

(322 citation statements)

References 32 publications

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“…The link between relaxation time and static correlation length derived by Montanari and Semerdjian [36], which we have somewhat heuristically extended and used in this paper, puts a bound on the contribution that can be attributed to a collective or "cooperative" activated mechanism driven by the growth of a static length scale. We have seen that this contribution stays rather modest in the dynamical range studied.…”

Section: Discussionmentioning

confidence: 77%

“…More recently, approaches that detect the growth in static correlations while staying clear of any specific proposal about local order, i.e., "order-agnostic" approaches, have been developed. Among these proposals, we note patch repetition lengths [30,31], length scales extracted from information theoretic analysis [32,33] or from finite-size studies of the configurational entropy [34], and other "point-to-set" correlation lengths [35][36][37][38][39][40].…”

Section: Introductionmentioning

confidence: 99%

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Decorrelation of the static and dynamic length scales in hard-sphere glass formers

Charbonneau

Tarjus

2013

Phys. Rev. E

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We show that, in the equilibrium phase of glass-forming hard-sphere fluids in three dimensions, the static length scales tentatively associated with the dynamical slowdown and the dynamical length characterizing spatial heterogeneities in the dynamics unambiguously decorrelate. The former grow at a much slower rate than the latter when density increases. This observation is valid for the dynamical range that is accessible to computer simulations, which roughly corresponds to that accessible in colloidal experiments. We also find that, in this same range, no one-to-one correspondence between relaxation time and point-to-set correlation length exists. These results point to the coexistence of several relaxation mechanisms in the dynamically accessible regime of three-dimensional hard-sphere glass formers.

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Section: Discussionmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Decorrelation of the static and dynamic length scales in hard-sphere glass formers

Charbonneau

Tarjus

2013

Phys. Rev. E

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“…In particular, it was measured in several numerical simulations [16][17][18][19][20] and shown to grow mildly in the (rather high) temperature regime investigated. Rigorous results have also strengthened its relevance: in [21] it was proven that if the relaxation time-scale, τ α , diverges in a superArrhenius way, either at finite temperature or at zero temperature, then ξ PTS has to diverge too, at least as fast as (T log τ ) 1/d (d being the spatial dimension). The definition of the point-to-set length is the following: take a typical equilibrium configuration, freeze the positions of all particles outside a sphere centered around a given point and study how the thermodynamics of the remaining particles, inside the sphere, is influenced by this amorphous boundary condition; ξ PTS is the smallest radius of the sphere at which the boundary has no longer any effect on the configuration at the center.…”

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confidence: 95%

Comparison of Static Length Scales Characterizing the Glass Transition

2013

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The dramatic dynamic slowing down associated with the glass transition is considered by many to be related to the existence of a static length scale that grows when temperature decreases. Defining, identifying and measuring such a length is a subtle and non-trivial problem. Recently, two proposals, based on very different insights regarding the relevant physics, were put forward. One approach is based on the point-to-set correlation technique and the other on the scale where the lowest eigenvalue of the Hessian matrix becomes sensitive to disorder. In this Letter we present numerical evidence that the two approaches might result in the same identical length scale. This provides further mutual support to their relevance and, at the same time, raise interesting theoretical questions, discussed in the conclusion, concerning the fundamental reason for their relationship.A natural assumption that prevails in the physics community regarding the glass transition is that together with the spectacular slowing down associated with this phenomenon there should be also a length scale that increases in accordance with the increased time scale [1]. Decades of research efforts were devoted to find such a length [2,3]. The discovery of dynamic heterogeneity in super-cooled liquids both in experiments and in theoretical studies [4] and the detailed analysis of the associated dynamical length-scale using multi-point correlation functions [5][6][7][8][9][10][11][12] revealed that more and more particles move in a correlated way approaching the glass transition. There is no doubt by now that there exists a growing dynamic length accompanying the glass transition. Although this is a new and important facet of the phenomenology of super-cooled liquids, it is still not clear to what extent dynamic correlations are the consequence or the primary origin of slow dynamics [11]. Another natural candidate for the dominant length-scale of glassy dynamics is a static length. Finding it remained an open problem for a long time. Recently, two very specific and very different methods to determine the elusive characteristic length scales had been proposed and implemented. In this Letter we review both methods and provide numerical indications that although the two methods are extremely different, they seem to lead to the same length scale. This provides further mutual support to their relevance and, at the same time, raises interesting theoretical questions concerning the fundamental reason for their relationship.The first method is based on the "point-to-set" (PTS) length [13,14] that will denote henceforth as ξ PTS . This length, originally introduced to characterize the real space structure of the so called "mosaic state" arising in the Random First Order Transition theory [13,15], allows one to probe the spatial extent of positional amorphous order. It has attracted a lot of attention recently. In particular, it was measured in several numerical simulations [16][17][18][19][20] and shown to grow mildly in the (rather high) temperature regime ...

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“…Inspired by critical phenomena, it is natural to expect that the slowing down of the dynamics is related to the vicinity of a thermodynamic phase transition, where some kind of long-range order would set in 11 . This is the spirit of different recent theories 4,9,[12][13][14] , but seems at odds with others 5,15 , at least at first sight.…”

mentioning

confidence: 99%

“…Furthermore, following suggestions based on RFOT 4,9,12 , it is natural to conjecture that we are probing the growth of positional amorphous order. Interestingly, in this scenario, all finite-point static correlations remain featureless, whereas the point-to-set correlations 11,26 …”

mentioning

confidence: 99%

Thermodynamic signature of growing amorphous order in glass-forming liquids

Biroli

Bouchaud

Cavagna³

et al. 2008

Nature Phys

346

468

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Supercooled liquids exhibit a pronounced slowdown of their dynamics on cooling 1 without showing any obvious structural or thermodynamic changes 2 . Several theories relate this slowdown to increasing spatial correlations 3-6 . However, no sign of this is seen in standard static correlation functions, despite indirect evidence from considering specific heat 7 and linear dielectric susceptibility 8 . Whereas the dynamic correlation function progressively becomes more non-exponential as the temperature is reduced, so far no similar signature has been found in static correlations that can distinguish qualitatively between a high-temperature and a deeply supercooled glass-forming liquid in equilibrium. Here, we show evidence of a qualitative thermodynamic signature that differentiates between the two. We show by numerical simulations with fixed boundary conditions that the influence of the boundary propagates into the bulk over increasing length scales on cooling. With the increase of this static correlation length, the influence of the boundary decays non-exponentially. Such long-range susceptibility to boundary conditions is expected within the random first-order theory 4,9,10 (RFOT) of the glass transition. However, a quantitative account of our numerical results requires a generalization of RFOT, taking into account surface tension fluctuations between states.Inspired by critical phenomena, it is natural to expect that the slowing down of the dynamics is related to the vicinity of a thermodynamic phase transition, where some kind of long-range order would set in 11 . This is the spirit of different recent theories 4,9,[12][13][14] , but seems at odds with others 5,15 , at least at first sight. In particular, the crucial physical mechanism at the root of random first-order theory 4 (RFOT) is the emergence of long-range amorphous order, the precise definition and quantitative characterization of which is however far from obvious. Dynamic heterogeneities 16 do show a growing dynamic correlation length accompanying the glass transition, both experimentally 17 and numerically 18 . This is certainly a first important step, but not sufficient to prune down-even at a qualitative level-different theories of the glass transition. In particular, it is not clear whether this phenomenon is due to an underlying static or purely dynamic phase transition.The approach followed here is based on the very definition of a thermodynamic phase transition, where the effect of boundary conditions becomes long-ranged. The problem is that for glasses there are no natural boundary conditions, because these should be as 'random' as the bulk amorphous states that they favour. A possible solution is to use equilibrium liquid configurations to define the boundary 19 . In the context of RFOT, this was suggested in ref. 9 (and further discussed in ref. 11), but the scope and some conclusions of this Gedankenexperiment are more general [19][20][21] . Starting from a given equilibrium configuration, we freeze the motion of all particles outside a ...

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Rigorous Inequalities Between Length and Time Scales in Glassy Systems

Cited by 225 publications

References 32 publications

Decorrelation of the static and dynamic length scales in hard-sphere glass formers

Decorrelation of the static and dynamic length scales in hard-sphere glass formers

Comparison of Static Length Scales Characterizing the Glass Transition

Thermodynamic signature of growing amorphous order in glass-forming liquids

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