A simple theory for the dynamics of mean-field-like models of   glass-forming fluids

Grzegorz Szamel

arXiv:1705.00036·cond-mat.dis-nn·October 18, 2017

A simple theory for the dynamics of mean-field-like models of glass-forming fluids

Grzegorz Szamel

PDF

1 Repo

TL;DR

This paper introduces a simple, physically motivated mean-field-like theory for the dynamics of glass-forming fluids, predicting an ergodicity-breaking transition aligned with the replica approach, and reproducing recent high-dimensional results.

Contribution

It offers a new, transparent theoretical framework for glass dynamics that can be tested via simulations and connects with existing replica and high-dimensional results.

Findings

01

Predicts an ergodicity-breaking transition in glass-forming fluids.

02

Aligns with the dynamic transition from the replica approach.

03

Reproduces results from high-dimensional exact calculations.

Abstract

We propose a simple theory for the dynamics of model glass-forming fluids, which should be solvable using a mean-field-like approach. The theory is based on transparent physical assumptions, which can be tested in computer simulations. The theory predicts an ergodicity-breaking transition that is identical to the so-called dynamic transition predicted within the replica approach. Thus, it can provide the missing dynamic component of the random first order transition framework. In the large-dimensional limit the theory reproduces the result of a recent exact calculation of Maimbourg et al. [PRL 116, 015902 (2016)]. Our approach provides an alternative, physically motivated derivation of this result.

Equations64

γ \dot{r}_{1} (t) = - \int_{0}^{t} M^{irr} (t - t^{'}) \dot{r}_{1} (t^{'}) d t^{'} + η_{1} (t) + ξ_{1} (t) .

γ \dot{r}_{1} (t) = - \int_{0}^{t} M^{irr} (t - t^{'}) \dot{r}_{1} (t^{'}) d t^{'} + η_{1} (t) + ξ_{1} (t) .

⟨ η_{i} (t) η_{j} (t^{'}) ⟩ = T δ_{ij} I M^{irr} (t - t^{'}),

⟨ η_{i} (t) η_{j} (t^{'}) ⟩ = T δ_{ij} I M^{irr} (t - t^{'}),

M^{irr} (t) = β ⟨ \hat{k} \cdot F_{1} e^{Ω^{irr} t} \hat{k} \cdot F_{1} ⟩ .

M^{irr} (t) = β ⟨ \hat{k} \cdot F_{1} e^{Ω^{irr} t} \hat{k} \cdot F_{1} ⟩ .

M^{irr} (t)

M^{irr} (t)

γ \dot{r}_{1} (t)

γ \dot{r}_{1} (t)

γ \dot{r}_{2} (t)

\tilde{γ} \dot{r} (t) = F (r (t)) - \int_{0}^{t} \tilde{M} (t - t^{'}) \dot{r} (t^{'}) d t^{'} + η (t) + ξ (t)

\tilde{γ} \dot{r} (t) = F (r (t)) - \int_{0}^{t} \tilde{M} (t - t^{'}) \dot{r} (t^{'}) d t^{'} + η (t) + ξ (t)

\tilde{M} (t) = \frac{1}{2} M^{irr} (t) = \frac{n β}{2} \int d r \hat{k} \cdot F (r (t)) \hat{k} \cdot F (r) g (r)

\tilde{M} (t) = \frac{1}{2} M^{irr} (t) = \frac{n β}{2} \int d r \hat{k} \cdot F (r (t)) \hat{k} \cdot F (r) g (r)

\tilde{M}_{EA} = \frac{n β}{2} \int d s P_{slow} (s) ⟨ \hat{k} \cdot F (r) ⟩_{s}^{2}

\tilde{M}_{EA} = \frac{n β}{2} \int d s P_{slow} (s) ⟨ \hat{k} \cdot F (r) ⟩_{s}^{2}

P_{slow} (s) = \int \frac{d r}{( 2 π T M ~ _{EA} ) ^{d /2}} e^{- β V (r) - \frac{( s - M ~ _{EA} r ) ^{2}}{2 T M ~ _{EA}}}

P_{slow} (s) = \int \frac{d r}{( 2 π T M ~ _{EA} ) ^{d /2}} e^{- β V (r) - \frac{( s - M ~ _{EA} r ) ^{2}}{2 T M ~ _{EA}}}

⟨ f (r) ⟩_{s} = \frac{\int d r e ^{- β (V (r) + \tilde{M}_{EA} r^{2} /2 - s \cdot r)} f ( r )}{\int d r e ^{- β (V (r) + \tilde{M}_{EA} r^{2} /2 - s \cdot r)}} .

⟨ f (r) ⟩_{s} = \frac{\int d r e ^{- β (V (r) + \tilde{M}_{EA} r^{2} /2 - s \cdot r)} f ( r )}{\int d r e ^{- β (V (r) + \tilde{M}_{EA} r^{2} /2 - s \cdot r)}} .

\tilde{γ} \partial_{t} r (t)

\tilde{γ} \partial_{t} r (t)

\overset{γ}{^} \dot{h} (t) = - w^{'} (h (t)) + \int_{0}^{t} M (t - t^{'}) \dot{h} (t^{'}) d t^{'} + η (t) + ξ (t)

\overset{γ}{^} \dot{h} (t) = - w^{'} (h (t)) + \int_{0}^{t} M (t - t^{'}) \dot{h} (t^{'}) d t^{'} + η (t) + ξ (t)

M (t) = \frac{β ϕ ^}{2} \int d h F (h) F (h (t)) e^{- β w (h)},

M (t) = \frac{β ϕ ^}{2} \int d h F (h) F (h (t)) e^{- β w (h)},

\tilde{M}_{EA}^{mct}

\tilde{M}_{EA}^{mct}

η (t) = η^{f} (t) + \overset{ˉ}{η} (t)

η (t) = η^{f} (t) + \overset{ˉ}{η} (t)

⟨ η^{f} (t) η^{f} (t^{'}) ⟩ = I T \tilde{M}^{f} (t - t^{'}) and ⟨ \overset{ˉ}{η} (t) \overset{ˉ}{η} (t^{'}) ⟩ = I T \tilde{M}^{s} (t - t^{'}),

⟨ η^{f} (t) η^{f} (t^{'}) ⟩ = I T \tilde{M}^{f} (t - t^{'}) and ⟨ \overset{ˉ}{η} (t) \overset{ˉ}{η} (t^{'}) ⟩ = I T \tilde{M}^{s} (t - t^{'}),

\tilde{γ} \dot{r} (t)

\tilde{γ} \dot{r} (t)

P (r ∣ s) = \frac{e ^{- β H (r ∣ s)}}{Z _{s}}

P (r ∣ s) = \frac{e ^{- β H (r ∣ s)}}{Z _{s}}

H (r ∣ s) = V (r) + \frac{1}{2} \tilde{M}_{EA} r^{2} - s \cdot r

H (r ∣ s) = V (r) + \frac{1}{2} \tilde{M}_{EA} r^{2} - s \cdot r

Z_{s} = \int d r exp (- β (V (r) + \frac{1}{2} \tilde{M}_{EA} r^{2} - s \cdot r)) .

Z_{s} = \int d r exp (- β (V (r) + \frac{1}{2} \tilde{M}_{EA} r^{2} - s \cdot r)) .

⟨ F (r) ⟩_{s} = \frac{1}{Z _{s}} \int d r e^{- β H (r ∣ s)} F (r),

⟨ F (r) ⟩_{s} = \frac{1}{Z _{s}} \int d r e^{- β H (r ∣ s)} F (r),

\tilde{M}_{EA}

\tilde{M}_{EA}

\tilde{M}_{EA} = \frac{n T}{2 d} \int \frac{d s e ^{- \frac{s ^{2}}{2 T M ~ _{EA}}}}{( 2 π T M ~ _{EA} ) ^{d /2}} \frac{\int d r _{1} δ ( r _{1} - σ ) e ^{- β (\tilde{M}_{EA} r_{1}^{2} /2 - s \cdot r_{1})} \int d r _{2} δ ( r _{2} - σ ) e ^{- β (\tilde{M}_{EA} r_{2}^{2} /2 - s \cdot r_{2})} r ^ _{1} \cdot r ^ _{2}}{\int d r _{3} θ ( r _{3} - σ ) e ^{- β (\tilde{M}_{EA} r_{3}^{2} /2 - s \cdot r_{3})}}

\tilde{M}_{EA} = \frac{n T}{2 d} \int \frac{d s e ^{- \frac{s ^{2}}{2 T M ~ _{EA}}}}{( 2 π T M ~ _{EA} ) ^{d /2}} \frac{\int d r _{1} δ ( r _{1} - σ ) e ^{- β (\tilde{M}_{EA} r_{1}^{2} /2 - s \cdot r_{1})} \int d r _{2} δ ( r _{2} - σ ) e ^{- β (\tilde{M}_{EA} r_{2}^{2} /2 - s \cdot r_{2})} r ^ _{1} \cdot r ^ _{2}}{\int d r _{3} θ ( r _{3} - σ ) e ^{- β (\tilde{M}_{EA} r_{3}^{2} /2 - s \cdot r_{3})}}

1 = \frac{2 ^{d} ϕ}{d} \frac{A}{1 - m} \frac{\partial G ( m , A )}{\partial A}

1 = \frac{2 ^{d} ϕ}{d} \frac{A}{1 - m} \frac{\partial G ( m , A )}{\partial A}

G (m, A) = \frac{1}{V _{d}} \int d r θ (r - σ) \int d r_{1} f_{2 A}^{G} (r_{1}) (q_{A} (r - r_{1})^{m - 1} - 1)

G (m, A) = \frac{1}{V _{d}} \int d r θ (r - σ) \int d r_{1} f_{2 A}^{G} (r_{1}) (q_{A} (r - r_{1})^{m - 1} - 1)

q_{A} (r) = \int d r_{2} f_{2 A}^{G} (r_{2}) θ (∣ r - r_{2} ∣ - σ),

q_{A} (r) = \int d r_{2} f_{2 A}^{G} (r_{2}) θ (∣ r - r_{2} ∣ - σ),

f_{A}^{G} (r) = \frac{e ^{- r^{2} /2 A}}{( 2 π A ) ^{d /2}} .

f_{A}^{G} (r) = \frac{e ^{- r^{2} /2 A}}{( 2 π A ) ^{d /2}} .

A = \frac{T}{2 M ~ _{EA}} .

A = \frac{T}{2 M ~ _{EA}} .

m \to 1 lim \frac{2 ^{d} ϕ}{d} \frac{A}{1 - m} \frac{\partial G ( m , A )}{\partial A} = - \frac{n}{2 d} \int d r q_{A}^{- 1} (r) \int d r_{1} \int d r_{2} θ (∣ r + r_{1} ∣ - σ) f_{2 A}^{G} (r_{1}) f_{2 A}^{G} (r_{2} - r) δ (r_{2} - σ) \hat{r}_{2} \cdot (r_{2} - r)

m \to 1 lim \frac{2 ^{d} ϕ}{d} \frac{A}{1 - m} \frac{\partial G ( m , A )}{\partial A} = - \frac{n}{2 d} \int d r q_{A}^{- 1} (r) \int d r_{1} \int d r_{2} θ (∣ r + r_{1} ∣ - σ) f_{2 A}^{G} (r_{1}) f_{2 A}^{G} (r_{2} - r) δ (r_{2} - σ) \hat{r}_{2} \cdot (r_{2} - r)

+ \frac{n}{2 d} \int d r q_{A}^{- 1} (r) \int d r_{1} \int d r_{2} θ (∣ r + r_{1} ∣ - σ) f_{2 A}^{G} (r_{1}) f_{2 A}^{G} (r_{2} - r) δ (r_{2} - σ) \hat{r}_{2} \cdot r_{1}

Peer Reviews

No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.

Code & Models

Repositories

mbaityje/STRUCTURAL-GLASS
none

Videos

No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.

Full text

A simple theory for the dynamics of mean-field-like models of

glass-forming fluids

Grzegorz Szamel

Department of Chemistry, Colorado State University, Fort Collins, CO 80523

Abstract

We propose a simple theory for the dynamics of model glass-forming fluids, which should be solvable using a mean-field-like approach. The theory is based on transparent physical assumptions, which can be tested in computer simulations. The theory predicts an ergodicity-breaking transition that is identical to the so-called dynamic transition predicted within the replica approach. Thus, it can provide the missing dynamic component of the random first order transition framework. In the large-dimensional limit the theory reproduces the result of a recent exact calculation of Maimbourg et al. [PRL 116, 015902 (2016)]. Our approach provides an alternative, physically motivated derivation of this result.

In the last decade, the static component of the general theoretical framework known as the random first order transition (RFOT) theory KWPRA1987 ; KTPRB1987 ; KWPRB1987 was developed into a consistent, albeit arguably mean-field-like, static description of the glass transition ParisiZamponiRMP ; BerthierBiroliRMP ; MezardParisiWolLubbook . This has been achieved by a combination of the generalization of the replica approach to amorphous systems without quenched disorder Monasson with the theoretical apparatus of the liquid state theory HansenMcDonald . More recently, it has been realized that in the limit of large spatial dimension the mean-field approach becomes exact and the complete theory of the glass (and jamming) transition in the infinite-dimensional hard-sphere system has been worked out largedCKPUZNC ; largedCKPUZAR .

On the other hand, the dynamic component of the RFOT approach has not been advanced to the same degree. This is a bit disappointing since the glass transition, as observed either in a laboratory or a computer experiment, manifests itself most clearly through the enormous slowing down of the dynamics. While within the original $p$ -spin version of the RFOT approach KTPRB1987 (and within the very simplified hard-sphere calculation of Ref. KWPRA1987 ) static and dynamic components of the theory were fully consistent, there is no finite-dimensional dynamic theory consistent with the advanced version of the replica-based static approach. In contrast, in the limit of large spatial dimension the dynamics of the hard-sphere system has been solved MaimbourgPRL . More precisely, the many-body problem has been reduced to a one-dimensional stochastic equation with colored noise, which is determined self-consistently. At high density, this equation predicts an ergodicity-breaking transition which is fully consistent with the so-called dynamic transition predicted by the large-dimensional replica calculation. Thus, the calculation of Maimbourg et al. suggests that it should be possible to come up with dynamical theories that agree with static approaches. Unfortunately, the physical content of the Maimbourg et al. result is a bit obscured by a rather long derivation.

One should note at this point that until the formulation of the static replica approach, the mode-coupling theory of the glass transition Goetzebook was the most successful quantitative description of the glass transition and for a long time was considered to be the dynamic mean-field theory of this transition (indeed, it was featured in this role in the original RFOT theory papers). However, it was discovered in the last decade SchmidSchillingPRE ; IkedaMiyazakiPRL2010 that in the large-dimensional limit the ergodicity-breaking transition predicted by the mode-coupling theory is different from the dynamic transition of the replica approach (which becomes exact in the large-dimensional limit).

The somewhat uncertain status of the mode-coupling theory resulting from its inadequacy in the large-dimensional limit JPBJCCM is disappointing in view of the fact that attempts to go beyond a mean-field-like description of the dynamics usually start from the mode-coupling theory KSS ; SFH2013 ; Rizzo2014 . Parenthetically, we should note that most attempts to go beyond a mean-field-like static description of the glass transition use lattice-based effective Hamiltonians rather than particle-based models CBTT2011 ; BCTT2014 .

Our goal in this Letter is to present a simple theory for glassy dynamics, which predicts ergodicity-breaking transitions consistent with dynamic transitions predicted by the replica approach. This theory can become the missing dynamic component of the RFOT approach. The theory is based on transparent physical assumptions. The most important assumption is that there should be no “loops” in the dynamics (this notion and its consequences are discussed in the following). In principle, this assumption makes the theory applicable only to mean-field-like models of glass-forming fluids. In particular, we show that the large-dimensional limit of our theory coincides with the exact result derived by Maimbourg et al. In finite dimensions the situation is a bit more complex. Additional assumptions of the theory, the most prominent one being the Gaussian character of the single-particle motion, make it only an approximate description of the dynamics of finite-dimensional mean-field-like models. On the other hand, analogous assumptions are used in the replica theory description of finite-dimensional mean-field-like models ParisiZamponiRMP (although, in principle, one can avoid these assumptions using the cavity approach of Mézard et al. MPTZ ). The result is that the ergodicity-breaking transition predicted by our theory for mean-field-like models in finite dimensions is the same as the dynamic transition predicted by the replica description.

To make our considerations more specific, we use the Mari-Krzakala-Kurchan model MKK . We consider $M$ particles in $d$ dimensional space. Any given particle interacts, via a spherically symmetric potential $V(r)$ , only with $N$ other particles, with $N\ll M$ . The network specifying inter-particle interactions forms a quenched random tree-like graph. This graph does not have “loops” and thus dynamical events in which two interacting particles interact with the same third particle are absent. This fact makes the model solvable via a mean-field-like approach (at least in principle). In particular, for this model the pair correlation function is equal to the Boltzmann factor, $g(r)=e^{-\beta V(r)}$ , with $\beta=1/T$ (we use a system of units in which $k_{B}=1$ ). At the initial time the particles are distributed according to the canonical ensemble. They evolve with Brownian dynamics; thus, each particle is subjected to the inter-particle interaction and a random Gaussian white noise. We note that in the large-dimensional limit the present model system and any system with short-range interactions are identical. This is due to the fact that in large dimensions the probability of “loops” is vanishingly small for geometric reasons.

The first (although not the most important) assumption of our approach is concerned with the description of the single-particle motion. Specifically, we assume that the motion of one selected (tagged) particle in the fluid of interacting particles can be described by the following generalized Langevin equation with Gaussian noise,

[TABLE]

Here, $\gamma$ is the friction coefficient of an isolated particle and $M^{\text{irr}}(t)$ is the irreducible memory function describing the average response of the other particles of the fluid to the motion of the tagged particle. The memory function is essentially a friction kernel, i.e. the first term at the right-hand-side of Eq. (1) is the internal friction force experienced by the tagged particle due to the presence of other particles. Next, $\boldsymbol{\eta}_{i}(t)$ in Eq. (1) is a Gaussian colored noise describing the fluctuating force acting on particle $i$ originating from the presence of the other particles. In equilibrium, the noise should be related to the memory function by a fluctuation-dissipation relation,

[TABLE]

where $\boldsymbol{I}$ is the unit tensor. Finally, $\boldsymbol{\xi}_{i}(t)$ in Eq. (1) is a Gaussian white noise acting on the particle $i$ , with autocorrelation function $\left<\boldsymbol{\xi}_{i}(t)\boldsymbol{\xi}_{j}(t^{\prime})\right>=2\gamma T\delta_{ij}\boldsymbol{I}\delta(t-t^{\prime})$ .

The generalized Langevin equation (1) can be justified as follows. Using the standard projection operator considerations one can derive an exact but formal equation for the time evolution of the tagged density auto-correlation function, which involves a wave-vector dependent irreducible memory function. From the evolution equation one can derive an equation of motion for the tagged particle’s mean-square displacement. The latter equation involves the zero-wave-vector limit of the irreducible memory function. If the particle’s motion can be described by a Gaussian stochastic process, then one can deduce Eq. (1) from the time evolution of the tagged particle’s mean-square displacement. It is possible that an exact but formal equation similar to Eq. (1) could be derived, but with some kind of a position dependent irreducible memory function and non-Gaussian noise.

The second, somewhat more technical, assumption of our approach is concerned with the irreducible memory function. One can derive an exact but formal expression for this function as an autocorrelation function of the total force acting on the tagged particle evolving with the so-called irreducible dynamics CHess ,

[TABLE]

Here $\hat{\mathbf{k}}$ is a unit vector, $\mathbf{F}_{1}=\sum_{i>1}\mathbf{F}_{1i}$ is the total force acting on the tagged particle (the summation extends over the particles that particle $1$ interacts with) and $\Omega^{\text{irr}}$ is the irreducible evolution operator CHess (note that our definition of the memory function includes an additional factor $\beta$ compared with the definition of Maimbourg et al.). Here, following Maimbourg et al., we will assume that the memory function can be obtained from the pair force evolving with the standard dynamics,

[TABLE]

where $\Omega$ is the standard Smoluchowski operator describing the motion of interacting Brownian particles.

The validity of assumption (4) can be checked almost directly. One could evaluate the right-hand-side of Eq. (4) in a Brownian dynamics computer simulation and then calculate the time-dependence of the mean-squared displacement from this approximate irreducible memory function. This calculated approximate mean-square displacement could then be compared to the mean-squared displacement measured in the same simulation.

It is clear that in general Eq. (4) is only an approximation. For example, for the finite-dimensional model with short-range interactions even the initial ( $t=0$ ) values of the exact memory function (3) and the approximation (4) are different. However, for a Mari-Krzakala-Kurchan model, due to the loop-less structure of the inter-particle interactions network, the initial values of these functions are the same. It follows from Maimbourg et al. that this is also true in the large-dimensional limit.

The second assumption, Eq. (4), suggests that one can evaluate the memory function by calculating the force between two selected particles, evolving with the standard dynamics, given that at the initial time these two particles were distributed according to probability distribution $\propto\hat{\mathbf{k}}\cdot\mathbf{F}_{12}g(r_{12})$ , where $g(r_{12})$ is the equilibrium pair correlation function.

Let us now consider the two-particle dynamics. The force acting on one of these particles consists of the following parts. The first part is the force due to the second particle. The second part describes the interaction with the other particles of the fluid, i.e. particles different from the interacting pair of particles. The latter force has the structure similar to the force acting on one particle, Eq. (1). It consists of an average part and fluctuating part. For a Mari-Krzakala-Kurchan model (and for any model in the large-dimensional limit) the force acting on the first particle and originating from other particles of the fluid should be independent of the state of the second particle (and vice versa). This is due to the fact that in these models the particles that interact with the first particle (and the particles that interact with these particles etc.) do not interact with the second particle. We have to emphasize here that the argument described in this paragraph relies on our most important assumption that the structure of the inter-particle interactions is such that “loops” are absent both in the statics and in the dynamics.

The argument formulated above leads us to assume the following equations of motion for the dynamics of the interacting pair particles,

[TABLE]

Obviously, to calculate the inter-particle force we only need the relative position, $\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}$ . The equation of motion for the relative position can be easily obtained from Eqs. (A simple theory for the dynamics of mean-field-like models of glass-forming fluids-A simple theory for the dynamics of mean-field-like models of glass-forming fluids),

[TABLE]

where $\tilde{\gamma}=\frac{1}{2}\gamma$ , $\tilde{M}(t)=\frac{1}{2}M^{\text{irr}}(t)$ , $\boldsymbol{\eta}(t)=\frac{1}{2}\left(\boldsymbol{\eta}_{1}(t)-\boldsymbol{\eta}_{2}(t)\right)$ , $\boldsymbol{\xi}(t)=\frac{1}{2}\left(\boldsymbol{\xi}_{1}(t)-\boldsymbol{\xi}_{2}(t)\right)$ and thus $\left<\boldsymbol{\xi}(t)\boldsymbol{\xi}(t^{\prime})\right>=2\tilde{\gamma}T\boldsymbol{I}\delta(t-t^{\prime})$ and $\left<\boldsymbol{\eta}(t)\boldsymbol{\eta}(t^{\prime})\right>=\boldsymbol{I}T\tilde{M}(t-t^{\prime})$ . We note that Eq. (7) is to be solved with the initial condition distributed according to $P(\mathbf{r})=n\hat{\mathbf{k}}\cdot\mathbf{F}(\mathbf{r})g(r)$ where $n=N/V$ is the number density of particles interacting with the tagged particle and $g(r)=e^{-\beta V(r)}$ is the pair correlation function for the Mari-Krzakala-Kurchan model. Then, the memory function is given by

[TABLE]

where $\mathbf{r}(t=0)=\mathbf{r}$ .

We note that even though the noise $\boldsymbol{\eta}(t)$ is Gaussian (since it is defined as a difference of two Gaussian noises), the process describing the pair dynamics, i.e. $\mathbf{r}(t)$ , is not Gaussian. This is due to the fact that the relation between these two processes, i.e. Eq. (7), is non-linear.

In principle, the self-consistent solution of Eq. (7) determines the irreducible memory function. At present, an analytic solution of this equation is not available. We can, however, show that at high enough density Eqs. (7-8) predict an ergodicity-breaking transition. To this end we can use the argument similar to that used by Maimbourg et al. (see Sec. V.A.1 of the supplementary information for Ref. MaimbourgPRL ).

We assume that at high enough density or low enough temperature a two-step relaxation process sets up, with both the mean-square displacement and the irreducible memory function being temporarily arrested around their respective plateau values. For times in the plateau region, both parts of the noise $\boldsymbol{\eta}(t)$ and the corresponding part of the memory function can be treated as adiabatically slow. It can be showed (see the Appendix) that the plateau value of the memory function, $\tilde{M}_{\text{EA}}$ , can be expressed as follows,

[TABLE]

where the distribution of the slow variable $\mathbf{s}$ (which is a linear combination of the slow part of the noise and the initial condition) reads

[TABLE]

and the conditional average for a given value of the slow variable is defined as

[TABLE]

We should note that the specific form of Eqs. (10-11) follows from the assumed Gaussian character of the noise.

Eq. (9) is a self-consistent equation for the plateau value of the memory function, $\tilde{M}_{\text{EA}}$ . A non-zero solution of this equation signals breaking of the ergodicity. It can be showed (see the Appendix) that Eq. (9) is equivalent to the equation determining the dynamic transition within the replica approach CJPZPNAS2014 . One should recall that the latter equation was derived using an assumption of a Gaussian shape of the “cage”, which corresponds to our assumption of the Gaussian character of the single-particle motion and of the noise.

Now, let us show that in the large-dimensional limit our theory is identical to the exact result of Maimbourg et al. First, we should recognize the fact that in the large-dimensional limit the relative motion of two particles that are interacting at the initial time (which is the process described by Eq. (7) with the initial condition distributed according to $P(\mathbf{r})=n\hat{\mathbf{k}}\cdot\mathbf{F}(\mathbf{r})g(r)$ ) proceeds predominantly along the original direction of the relative coordinate, i.e. along $\mathbf{r}(t=0)$ . In other words, in the large-dimensional limit $\partial_{t}\hat{\mathbf{r}}(t)$ can be neglected. This allows us to focus on the equation of motion for the inter-particle distance $r(t)\equiv|\mathbf{r}(t)|$ . Using Itô’s convention we obtain,

[TABLE]

where $F(r(t))=\hat{\mathbf{r}}(t)\cdot\mathbf{F}(\mathbf{r}(t))$ and the first term in the second line originates from Itô’s lemma. Since $\hat{\mathbf{r}}(t)\approx\hat{\mathbf{r}}(t^{\prime})$ , we can express the second term at the right-hand-side of Eq. (12) in terms of $r(t^{\prime})$ . Next, we use the scaling relationships introduced by Maimbourg et al., which can also be deduced from the large-dimensional limit of Eq. (9): $r(t)=\sigma(1+h(t)/d)$ , $\hat{\gamma}=\tilde{\gamma}\sigma^{2}/d^{2}=\gamma\sigma^{2}/(2d^{2})$ , $F(h)=\frac{\sigma}{d}F(r)$ and $M(t)=\frac{\sigma^{2}}{d^{2}}\tilde{M}(t)$ , with $\sigma$ being the particle diameter. Then, in the large-dimensional limit we get from Eq. (12) the following one dimensional stochastic equation for the “gap” $h(t)$ ,

[TABLE]

where the effective force is given by $-w^{\prime}(h)=F(h)+T$ and the noises $\eta$ and $\xi$ satisfy fluctuation-dissipation relations $\left<\eta(t)\eta(t^{\prime})\right>=TM(t-t^{\prime})$ and $\left<\xi(t)\xi(t^{\prime})\right>=2\hat{\gamma}T\delta(t-t^{\prime})$ . Finally, from the memory function expression (4) we can get an expression for the re-scaled function $M(t)$ ,

[TABLE]

where $h(t=0)=h$ and $\hat{\phi}$ is the rescaled volume fraction, $\hat{\phi}=n\mathcal{V}_{d}2^{d}/d$ , with $\mathcal{V}_{d}$ being the volume of a $d$ -dimensional sphere of radius $\sigma/2$ .

Eqs. (A simple theory for the dynamics of mean-field-like models of glass-forming fluids-14) are identical to the equations derived by Maimbourg et al. In particular, according to these equations, there exists an ergodicity-breaking transition at $\hat{\phi}_{d}=4.807$ .

Let us comment on the connection of the present simple approach and the mode-coupling theory. The most important approximation of the latter theory is the factorization approximation in which a four-point correlation function is replaced by a product of two-point correlation functions. In the present approach, the analogue of this approximation would be neglecting the inter-particle force term, $\mathbf{F}(\mathbf{r})$ , in the stochastic equation of motion for the relative position, Eq. (7). However, if one were just to neglect the force term in Eq. (7), the expression (8) for the memory function would give an infinite result. The way out is to incorporate one of the additional approximations of the mode-coupling theory Goetzebook and to combine discarding the force term with replacing the “bare” forces in the memory function expression (8) by renormalized forces given by the derivatives of the direct correlation function. In the present case of the Mari-Krzakala-Kurchan model the last step amounts to the replacement $\mathbf{F}(\mathbf{r})\rightarrow T\partial_{\mathbf{r}}e^{-\beta V(r)}$ in expression (8). It can be showed that this procedure results in the following self-consistent equation for the plateau value of the memory function, $\tilde{M}_{\text{EA}}^{\text{mct}}$ , where the superscript mct indicates a mode-coupling-like approximation,

[TABLE]

By rewriting the right-hand-side of this equation as an integral in reciprocal space we can show that this equation is identical to Eq. (4.6’) of Kirkpatrick and Wolynes KWPRA1987 (with additional factors of 2 in a couple of places). In the large-dimensional limit this equation predicts an ergodicity-breaking transition at $\hat{\phi}_{d}^{\text{mct}}=\sqrt{8\pi e}=8.265$ . This finding agrees perfectly with the numerical result of Ikeda and Miyazaki IkedaMiyazakiPRL2010 obtained by taking the large-dimensional limit of the standard mode-coupling equation for the non-ergodicity parameter and assuming a Gaussian form of this parameter. We note that the mode-coupling-like version of our theory predicts the ergodicity-breaking transition at a higher value of the rescaled volume fraction. This is reasonable in that the mode-coupling-like version neglects direct interaction between the two particles; replacing “bare” forces by renormalized ones only partially compensates for this fact.

In summary, we presented here a simple theory for the dynamics of models of structural glasses which should be solvable using a mean-field-like approach. The ergodicity-breaking transition predicted by our theory coincides with the dynamic transition of the replica approach. Thus, our theory provides the dynamic counterpart of the static replica approach.

In our theory, any given particle interacts explicitly only with one other particle of the system. Other interactions are accounted for by a combination of a friction force and a fluctuating force.

In the large-dimensional limit our theory reduces itself to the result of the exact calculation of Maimbourg et al. and thus it provides an alternative, physically motivated derivation of this result. In finite dimensions it suffers from the same problem as the replica approach, i.e. from an additional assumption of the Gaussian character of the single-particle dynamics. It would be very interesting to develop an analogue of a cavity approach of Mézard et al. which could possibly overcome this problem.

Our theory can shed some light onto somewhat abstract considerations of the replica approach. In particular, if we used the potential of the mean force $V^{\text{mf}}(r)=-T\ln g(r)$ rather than the true potential in Eqs. (7-8), the resulting ergodicity-breaking transition would coincide with the dynamic transition predicted by a recent version of the replica theory for a standard finite-dimensional system MangeatZamponi . Thus, as far as location of the dynamic transition is concerned, the latter approach approximates the standard hard-sphere system by a Mari-Krzakala-Kurchan system with particles interacting via a potential of mean force. Conversely, this observation means that by using the potential of the mean force, our simple theory can be made quantitatively accurate for the 3-dimensional hard sphere system, at least for the location of the dynamic transition.

Finally, we showed that a small modification of the present theory results in a mode-coupling-like approach with an additional assumption of Gaussian fluctuations. It would be interesting to start from a more standard mode-coupling-like approach by considering equations of motion for density fields and to develop an approximate theory by keeping only dynamical events which are included in the present approach. This might be another avenue that would allow us to avoid or relax the assumption of Gaussian fluctuations and thus arrive at a theory that both provides a reasonable, albeit mean-field-like, description of the dynamics of finite-dimensional systems and has the correct large-dimensional limit.

I benefited from stimulating discussions with Francesco Zamponi and Jorge Kurchan. I also thank both of them and Elijah Flenner for comments on the manuscript. I gratefully acknowledge the support of NSF Grant No. CHE 121340.

Appendix

I.1 I. Equation determining the ergodicity-breaking transition

Following Maimbourg et al. we assume that the noise $\boldsymbol{\xi}(t)$ , which describes the influence of other particles on the pair of interacting particles, can be split into a fast part and a slowly evolving part,

[TABLE]

with

[TABLE]

where $\tilde{M}^{f}$ evolves on the fast time scale $\tau_{f}$ and $\tilde{M}^{s}$ evolves on the slow time scale $\tau_{s}$ .

Then, we re-write the equation of motion for the relative position for times in the plateau region, using the above formulated assumption,

[TABLE]

where $\mathbf{s}=\tilde{M}_{\text{EA}}\mathbf{r}(0)+\bar{\boldsymbol{\eta}}(t)$ and $\tilde{M}_{\text{EA}}$ is the value of the slow part of the memory function $\tilde{M}^{s}$ in the plateau region.

At intermediate time scales (when $\mathbf{s}$ can be considered constant), the relative position relaxes to an “equilibrium” state for a given value of $\mathbf{s}$ . In this state $\mathbf{r}$ is distributed according to the probability distribution

[TABLE]

where

[TABLE]

and the normalization constant reads

[TABLE]

Now, to calculate the value of the slow part of the memory function we use Eq. (8) of the main text. We replace $\mathbf{F}(\mathbf{r}(t))$ by the force averaged with the conditional equilibrium distribution (I.4), i.e. by

[TABLE]

and then average over the slowly evolving noise $\bar{\boldsymbol{\eta}}$ , auxiliary variable $\mathbf{s}$ and the initial condition, which to avoid confusion we denote here by $\mathbf{r}_{0}$ . Thus,

[TABLE]

where $P_{\text{slow}}(\mathbf{s})$ is given by Eq. (10) of the main text.

II.2 II. Ergodicity-breaking transition vs. dynamic transition

of the replica approach

First, we re-write Eq. (9) of the main text for the hard-sphere system. To this end, we follow the standard procedure: we write $e^{-\beta V(r)}\mathbf{F}(\mathbf{r})$ as $T\partial_{\mathbf{r}}e^{-\beta V(r)}$ and then take the hard sphere limit. In this way we obtain the following self-consistent equation,

[TABLE]

We note that to get Eq. (II.1) we also averaged over arbitrary direction of unite vector $\hat{\mathbf{k}}$ .

Now, let us recall the equation which determines the dynamic transition of the replica approach. Eq. (S20) of Supporting Information of Ref. CJPZPNAS2014 reads

[TABLE]

where the right-hand side is to be calculated in the limit of $m\to 1$ . In Eq. (II.2), $\phi$ is the volume fraction, $\phi=n\mathcal{V}_{d}2^{-d}$ , with $\mathcal{V}_{d}$ being the volume of the sphere of radius $\sigma/2$ . We use Appendix C of Ref. ParisiZamponiRMP and write $G(m,A)$ as follows

[TABLE]

with

[TABLE]

and $f_{A}^{G}$ being a Gaussian distribution,

[TABLE]

The dynamic transition occurs at the lowest density at which Eq. (II.2) has a non-trivial solution for parameter $A$ . This parameter is proportional to the plateau value of the mean-square displacement, which in the dynamical approach is inversely proportional to the plateau value of the memory function. The final relation between $A$ and $\tilde{M}_{\text{EA}}$ has the following form,

[TABLE]

After some transformations, the $m\to 1$ limit of the right-hand-side of Eq. (II.2) can be written explicitly as follows,

[TABLE]

It can be showed that the first term at the right-hand-side of Eq. (II.2) vanishes. The second can be re-written as

[TABLE]

The resulting self-consistent equation for parameter $A$ thus reads

[TABLE]

which, with identification (II.6) and a simple change of variables can be written as Eq. (II.1). Thus, the ergodicity-breaking transition of our simple dynamical theory coincides with the dynamic transition of the replica approach.

Bibliography25

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1(1) T.R. Kirkpatrick and P.G. Wolynes, Phys. Rev. A 35 , 3072 (1987).
2(2) T.R. Kirkpatrick and D. Thirumalai, Phys. Rev. B 36 , 5388 (1987).
3(3) T.R. Kirkpatrick and P.G. Wolynes, Phys. Rev. B 36 , 8552 (1987).
4(4) G. Parisi and F. Zamponi, Rev. Mod. Phys. 82 , 789 (2010).
5(5) L. Berthier and G. Biroli, Rev. Mod. Phys. 83 , 587 (2011).
6(6) M. Mézard and G. Parisi, in Structural Glasses and Supercooled Liquids: Theory, Experiment, and Applications , edited by P.G. Wolynes and V. Lubchenko (Wiley, New York, 2012).
7(7) R. Monasson, Phys. Rev. Lett. 75 , 2847 (1995).
8(8) J.-P. Hansen and I.R. Mc Donald, Theory of Simple Liquids , (Academic, London, 1986).