Glassy dynamics

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Henrik Jeldtoft Jensen and Paolo Sibani (2007), Scholarpedia, 2(6):2030. doi:10.4249/scholarpedia.2030 revision #127071 [link to/cite this article]
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Curator: Henrik Jeldtoft Jensen

In solid-state physics, glassy dynamics designates the extremely slow dynamics observed in disordered systems below and slightly above the glass transition. Generally characterized as "relaxation", it comprises both the aging of quenched systems (relaxation into equilibrium) and fluctuations in a stationary state (relaxation in equilibrium). In a more general sense, the term glassy dynamics designates dynamical processes which are non-stationary on the time scales available to human observers. Such processes are often encountered in systems possessing, for whatever reason, a very large number of metastable configurations.


Structural Glasses

Structural glasses are formed by liquids that can be supercooled or quenched into an amorphous state. While the structure, as revealed by diffraction experiments, remains quite similar to the liquid state, the mobility of the atoms or molecules decreases by many orders of magnitude. This loss of mobility makes it impossible to reach thermodynamic equilibrium.

A glass forming liquid falls out of equilibrium in at least three distinct steps. Failure to distinguish these steps in connection with unclear language is a major source of confusion.

First, when supercooled below the melting point Tm, the liquid enters a non-equilibrium state with respect to crystallisation. Though the dichotomy equilibrium / non-equilibrium is generally considered a very fundamental one, it has hardly any importance for the physics of a supercooled liquid. Macroscopic and microscopic properties show no discontinuity whatsoever Tm. Only if nucleation really happens, the concurrence of a crystalline ground state must be taken into account.

Secondly, spectroscopic experiments test the linear response of the system to a small perturbation that oscillates with some frequency ω. In glass-forming systems one finds that the response changes from liquid-like to solid-like, depending on ω and T. The inverse of the cross-over frequency between these two regimes, τ(T)=ω-1, is the characteristic relaxation time of the system for a given spectroscopy at a given temperature. If τ>ω-1, fluctuations with frequency ω have fallen out of equilibrium.

Thirdly, the glass transition temperature Tg is a conventionally defined point where the system falls out of equilibrium with respect to the patience of a human observer: it is not viable (at least not routinely) to perform experiments on time scales longer than τ(Tg).

Structural glasses have been studied, experimentally and theoretically, for very long time. Dynamical approaches such as mode-coupling theory (first developed in other context by Kawasaki (1966)) have been developed in attempts to understand the change in dynamical response for temperatures higher than, and in the vicinity of the glass transition temperature, Bengtzelius (1984). The mode-coupling theory approximately treats the dynamics of the density fluctuations by analysing the time dependence of the correlations between the Fourier transformed density corrections \( \phi_q(t) =\frac{\langle \rho_q(t)^*\rho_q(0)\rangle}{\langle|\rho_q|^2\rangle} \) and is argued to be able to explain a large amount of experimental observations Kob (1997), Götze (1999).


The thermal equilibrium state of a glass below its glass transition temperature is experimentally unreachable. Considerable experimental and theoretical attention has therefore been focussed on the low temperature dynamics widely known as aging dynamics. Being surprisingly system independent, aging phenomena can be studied in different contexts, wherever convenient tools are available. Experimental and simulational probes usually involve linear response and autocorrelation functions, as well as the statistics of spontaneous fluctuations of physical properties, e.g. the energy. Aging was discovered in soft condensed matter, Struik et al. (1970), and later studied in great detail using magnetic glasses. Presently, aging phenomena are known in a large number of systems, including e.g. granular systems, Josserand et al. (2000), and colloids, Cipelletti et al. (2000).

Magnetic Glasses

Starting from the middle of the 1970ties, the discovery that some magnetic alloys, called spin glasses, possess magnetic properties dynamically similar to the out-of-equilibrium behavior observed in structural glasses lead to a surge in both theoretical and experimental studies of general aspects of glassy dynamics . The magnetic materials exhibit slow glassy dynamics because of the fixed or `quenched' random interactions between their microscopic magnetic moments. These random interactions stem from the fixed and random positions of the magnetic moments in the alloy, and prevent the moments to align in a preferred direction. Theoretical work on spin glasses initially focussed on the thermal equilibrium properties of the spin-glass phase, Hertz and Fischer (1991). The orientations of the spins in the spin-glass phase are random in space, and the average macroscopic moment is therefore zero at all temperatures. The subtle order of the spin-glass phase lies in the strong correlation present between the orientations of different spin. The true nature of the ground state and low energy states of a spin glass, and the related issue of whether an external magnetic field would destroy the spin-glass order or not have been the subjects of a prolonged debate. For Ising spin glasses with short range interactions, the 'droplet picture' of Fisher and Huse (1987), posits the existence of two ground states, which map into each other under a global sign reversal. The spin-glass order is destroyed by an arbitrary small external field. In contrast, the Parisi picture, Parisi (1979) of the spin glass, which stems from the analysis of a mean field model, posits the existence of a multitude of ground states, which are not related by symmetry operations. The phase diagram includes a spin-glass phase in finite external magnetic fields. A heated discussion has unfolded over almost three decades over the relevance of these two descriptions for real spin-glasses, Mattsson (1995). Magnetic linear response experiments in spin glasses have been instrumental in elucidating many fascinating aspects of aging dynamics. The so-called memory and rejuvenation effects, Vincent (1996), Jonason (1998), which are observed by applying small temperature variations to an aging system reveal the complex multi-scale structure characterizing spin-glass dynamics. The low temperature dynamics of spin-glasses and a host of other glassy material is mainly related to metastability, and only weakly reflects, if at all, the properties of the equilibrium state. In spin-glasses metastability is rooted in frustration: The individual magnetic moments experience many counter acting interactions that hinder them from quickly relaxing into a configuration that minimizes the thermodynamic free energy. This creates a large number of metastable regions in configurations space, variously called traps, valleys, or pockets of states. Within a metastable state, physical quantities e.g. the magnetization and energy, fluctuate reversibly around a constant value. The drift part of the dynamics is associated to shifts from one metastable state to a different one. The co-existence of equilibrium fluctuations with a slow non-equilibrium drift. or aging, is a general property of glassy dynamics at low temperatures.

Characteristic temperatures for glassy dynamics

A number of different temperatures, defined either empirically, or by mathematical extrapolations, is associated to glassy materials. Conventionally, the glass temperature \(T_g\) is defined as the temperature where the viscosity of a glass forming liquid exceeds \(\eta = 10^{13}\) Poise. For \( T < T_g\ ,\) thermal relaxation requires the collective re-organization of many degrees of freedom, resulting in non-ergodic behavior. The freezing of the motion of individual particles is revealed by a characteristic loss of configurational entropy.

Glass-formers are usually classified Angell (1985) as strong or fragile, according to how their viscosity behaves as \(T\) approaches \(T_g\) from above. In the so-called strong glass formers, the viscosity and the associated characteristic relaxation time \(\tau_R\) grow with \(T\) in an Arrhenius fashion. e.g. \( \tau_R \propto \exp(B/T)\ ,\) where \(B\) is a constant. Fragile glass formers follow the Vogel-Fulcher law \( \tau_R \propto \exp(B/(T-T_0)\ ,\) where the Vogel temperature \(T_0 < T_g\) is yet a new parameter.

With experiments limited to temperatures well above \(T_0\ ,\) the physical meaning of \(T_0\) remains somewhat unclear. A third parameter, the Kauzmann temperature, \(T_K < T_g\) is defined using a thermodynamical criterium: An extrapolation of the smooth T dependence of the entropy of the supercooled liquid below \(T_g\) crosses the entropy of the crystalline phase at \(T=T_K\ .\) The Vogel temperature and the Kauzmann temperature are widely believed to coincide. Last but not least, the aging dynamics below \(T_g\) is often described in terms of a so-called effective temperature. The off-equilibrium nature of aging is often recognized by the violation of one of the cornerstones of equilibrium statistical mechanics: the fluctuation-dissipation theorem. This theorem relates the autocorrelation of equilibrium fluctuations in an unperturbed system to the average response to a small external perturbation. In thermal equilibrium, fluctuations are time homogeneous and the autocorrelation \(C(\tau)\) and the response function \(R(\tau)\) depend on a single time variable, e.g. the time\(\tau = t-t_w\) elapsed from the instant t_w at which the field is applied.

By contrast, in glassy dynamics both quantities separately depend on both \(t_w\) and on \(t\ ,\) the time elapsed since the initial quench. Often used to characterize the out-of equilibrium dynamics are the concepts of fluctuation-dissipation ratio \(X\) and effective temperature \(T_{eff}\ ,\) Cuglliandolo (1997). These enter a generalization of the fluctuation-dissipation theorem \(T R(t,t_w)=X(t,t_w) \frac{\partial C(t,t_w)}{\partial t} \ ,\) where \(T\) is the temperature,Cugliandolo et al (1997). The fluctuation-dissipation ratio\( X(t,t_w)\) can be different from the equilibrium value \(X(t,t_w)=1\) and the time dependent effective temperatures is given by \(T(t,t_w)_{eff}\propto T/X(t,t_w)\ .\) The effective temperature can be calculated in the asymptotic limit of \( t\) and \(t_w\) diverging to infinity, with the ratio of the two quantities kept constant. The experimental relevance of effective temperatures remains however unclear.

Paradigmatic Relevance of Glassy Dynamics

Glassy dynamics has now been observed in very different systems, including non-thermal systems as granular materials and even non-physical systems as traffic flow and models of biological evolution. The possible relevance of glassy dynamics to adaptive immune response was suggested by Parisi (1990), and was more recently taken up by Sun et al. (2005). All glassy systems seem to involve a type of frustration, i.e., competing interactions make it difficult or impossible to reach an optimal, and stationary, state. For this very reason, the nature of the true stationary state becomes largely irrelevant for the dynamics. The frustration may often be of energetic nature, e.g. competing bonds between components, or entropic, as in jamming, a phenomenon similar to an ordinary traffic jam, where the motion of individual components becomes contingent on large scale collective rearrangements of surrounding components. In all cases, the system becomes trapped in long lived metastable states.

Figure 1: The dynamics of a complex system can be qualitatively summarised by considering the relation between time, configuration and fickleness. By fickleness we simply mean some relevant measure of stability or resilience. The smaller the fickleness value (i.e. the lower the value is along the z-axis), the more stable the system becomes. The long time dynamics consists of a slow evolution in the form of jumps, or quakes, from one metastable configuration to the next, as indicated by the sequence of ever deeper wells, or valleys, at the left of the figure. The quakes are only seen when the system is observed over many decades of time, hence the logarithmic time axis. The dynamics between the quakes is represented by the magnification shown on the right. On a linear (short) time scale, the system undergoes smaller jumps between sub-valleys within a single main valley. Short time dynamics slightly improves the stability of the system as indicated by the decrease of the system’s fickleness with time. The quakes have a similar effect on a logarithmic time scale, as indicated by the deepening of the valleys on the left of the figure.

Metastability in glassy system shows itself through the presence of a quasi-stationary fluctuation regime. In model simulation it is sometimes also possible to map out local energy minima configurations, or inherent states, Stillinger and Weber (1983) and their basins of attraction. The connection between the topography of the energy landscape and the dynamics of the system is of considerable theoretical interest: E.g. hierarchical models, Sibani and Hoffmann (1989) and Sibani et al. (1993) are coarse grained descriptions of configuration space which capture several aspects of aging dynamics, as they naturally give rise to the multitude of time scales implied by slow relaxation. The very popular model of Bouchaud (1992) builds on the idea that traps lack a characteristic residence time, i.e. the release from a trap is considered to be a slow process. A fundamental problems which all theories must face is how to seamlessly meld configuration space aspects, e.g. traps or metastable basins, and real space properties such as correlated domains, or gages, which characterize glassy systems with short range interactions. Recent experimental input on this issue is provided by intermittency studies of fluctuations in glassy systems, Buisson et al.(2003), which demonstrate that large intermittent fluctuations are responsible fro the deviations from equilibrium statistics. It was suggested, Sibani and Dall(2003) that abrupt and irreversible moves from one metastable configuration to another, so called quakes, are a a result of record sized fluctuations. While in a metastable configuration, fluctuations are small, reversible and Gaussianly distributed with zero average. The assumption that the metastable attractors typically selected by the glassy dynamics have marginally increasing stability, Sibani and Littlewood (1992), means that a fluctuation bigger than any previously occurred fluctuation, i.e. a record-sized fluctuation, can induce a quake. Quakes lead to entrenchment into gradually more stable configurations, and carry the average drift of the dynamics. These properties are experimentally verifiable using fluctuation data from mesoscopic system, e.g. the time series of the quake events and/or the Probability Distribution Function of the fluctuating quantity of interest, e.g. the energy or the linear response, Sibani et al. (2006).

The broad paradigmatic relevance of glassy dynamics was studied by Anderson et al. (2004). Here the long time relaxation of very different systems, magnetic relaxation of superconductors and models of macro-evolution , was analysed by method previously applied to spin glasses. An analysis that might turn out to be applicable to relaxation in complex systems in general.


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Attractor, Dynamical Systems

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