Exactly solvable models
[0.15truecm] of growing interfaces and lattice gases:
[0.16truecm] the Arcetri models, ageing and logarithmic sub-ageing
Xavier Durang111e-mail: firstname.lastname@example.org and Malte Henkel222e-mail: email@example.com
School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea
Department of Physics, University of Seoul, Seoul 02504, Korea
Rechnergestützte Physik der Werkstoffe, Institut für Baustoffe (IfB),
ETH Zürich, Stefano-Franscini-Platz 3, CH – 8093 Zürich, Switzerland
Groupe de Physique Statistique, Département de Physique de la Matière et des Matériaux, Institut Jean Lamour (CNRS UMR 7198), Université de Lorraine Nancy, B.P. 70239,
F – 54506 Vandœuvre lès Nancy Cedex, France333permanent address;
after 1 of Januar 2018: Laboratoire de Physique et Chimie Théoriques (CNRS UMR), Université de Lorraine Nancy, B.P. 70239, F – 54506 Vandœuvre-lès Nancy Cedex, France
Centro de Física Teórica e Computacional, Universidade de Lisboa,
P–1749-016 Lisboa, Portugal
Motivated by an analogy with the spherical model of a ferromagnet, the three Arcetri models are defined. They present new universality classes, either for the growth of interfaces, or else for lattice gases. They are distinct from the common Edwards-Wilkinson and Kardar-Parisi-Zhang universality classes. Their non-equilibrium evolution can be studied from the exact computation of their two-time correlators and responses. The first model, in both interpretations, has a critical point in any dimension and shows simple ageing at and below criticality. The exact universal exponents are found. The second and third model are solved at zero temperature, in one dimension, where both show logarithmic sub-ageing, of which several distinct types are identified. Physically, the second model describes a lattice gas and the third model interface growth. A clear physical picture on the subsequent time- and length-scales of the sub-ageing process emerges.
PACS numbers: 05.40.-a, 05.70.Ln, 81.10.Aj, 02.50.-r, 68.43.De
The physics of the growth of interfaces is a paradigmatic example of the emergence of non-equilibrium cooperative phenomena, with widespread applications in domains as different as deposition of atoms on a surface, solidification, flame propagation, population dynamics, crack propagation, chemical reaction fronts or the growth of cell colonies [3, 32, 53, 52, 17, 81, 77, 82]. Several universal growth and roughness exponents characterise the morphology of the growing interface and the time-dependent properties are quite analogous to phenomena encountered in the physical ageing in glassy and non-glassy systems [20, 37, 78]. Several universality classes of interface growth have been identified, the best-known of these are characterised in terms of stochastic equations for the height profile
where is the spatial gradient, is a centred gaussian white noise, with covariance
and are material-dependent constants.
While the exact solution of the ew-equation is straightforward, extracting the long-distance and/or long-time properties of interfaces in the kpz-class is considerably more difficult and several aspects of the problem still remain unresolved. Remarkable progress has been achieved in recent years on the exact solution of the kpz-equation in dimension. In particular, several spatial correlators have been found exactly and a deep relationship of the probability distribution of the fluctuation with the extremal value statistics of the largest eigenvalue of random matrices has been derived, see [69, 12, 13, 43, 33, 34, 48]. Very remarkably, these mathematical results could be confirmed experimentally, in several physically distinct systems [73, 40, 74, 75, 83, 36, 35, 41, 2, 76]. Still, this impressive progress seems to rely on specific properties of the one-dimensional case. Therefore, one might wonder if further classes of exactly solvable models of interface growth could be defined, distinct from both the ew- as well as the kpz-universality class, and what physical insight the study of such models might provide.
Indeed, a new class of models can be defined, with the help of some inspiration from the definition of the well-studied spherical model of a ferromagnet [7, 57]. Therein, the traditional Ising spins , attached to the sites of a lattice with sites, are replaced by continuous spins and subject to the ‘spherical constraint’ . A conventional nearest-neighbour interaction leads to an exactly solvable model, which undergoes a non-trivial phase transition in dimensions [7, 44]. The relaxational properties can be likewise analysed exactly, see e.g. [67, 14, 18, 19, 29, 27, 65, 22, 26, 30]. In order to identify an analogy with growing interfaces, we restrict here to dimensions for simplicity. Consider a lattice representation of the kpz-class where the height differences between two nearest neighbours obey the so-called rsos constraint . It is well-established that in the continuum limit this model is described by the kpz-equation [3, 32, 77], see  for a rigorous derivation. The dynamic deposition rule is sketched in figure 1, which makes it clear that in this kind of lattice model, the slopes should be considered as the analogues of the Ising spins in ferromagnets. For the slopes, in the continuum limit, from the kpz-equation follows the (noisy) Burgers equation  (for a discrete analogue, see )
A ‘spherical model variant’ of the kpz-universality class now stipulates to relax the rsos-constraints to a ‘spherical constraint’ .111An old observation by Oono and Puri  gives additional motivation: treating the Allen-Cahn equation of phase-ordering, after a quench to , along the lines of the celebrated Ohta-Jasnow-Kawasaki approximation, but for a finite thickness of the domain boundaries, leads to a kinetic equation in the universality class of the spherical model. However, for growing interfaces several equivalent descriptions can give rise to several new models, which may or may not be in the same universality class. Heuristically, the following possibilities may occur:
1. One may start from the Burgers equation and replace its non-linearity as follows
with a Lagrange multiplier which might be seen as some kind of ‘averaged curvature’ of the interface. Its value is determined by the mean spherical constraint222In this section, the average is understood to be taken over both the ‘thermal’ as well as the ‘initial’ noise. . This is the ‘first Arcetri model’, defined333The name comes from the location of the Galileo Galilei Institute of Physics, where this model was conceived. and analysed in .444It can be shown that for sufficiently long times, whenever . In any dimension , there is a ‘critical temperature’ such that long-range correlations build up for . At the critical point , the interface is rough for and is smooth for . For , the interface is always rough. The model is also related to the gaps in the spectra of random matrices  and to the spherical spin glass .
2. An alternative way to treat the Burgers equations might proceed as follows
where the Lagrange multiplier might now be viewed as some kind of ‘averaged slope’. Its value is again determined by the constraint . This would define a ‘second Arcetri model’.
3. Finally, we might have started directly from the kpz equation
where might again be interpreted as an ‘averaged slope’ and will be found from a constraint . This would be a ‘third Arcetri model’.
However, such a simplistic procedure would lead to undesirable properties of the height and slope profiles in the stationary state, as well as to internal inconsistencies. We shall therefore reconsider this correspondence carefully in section 2, where the precise definitions of the second and third Arcetri model will be given.
In one spatial dimension, the slope profile has an interesting relationship with the dynamics of interacting particles. To see this, write the slope as , where denotes the particle-density at time and position . In the kpz universality class, when on the lattice the rsos-constraint holds, denote by an occupied site with and by an empty site with . Then the only admissible reaction between neighbouring sites is the directed jump . The stochastic process described by these interacting particles is a totally asymmetric exclusion process (tasep), see e.g. [56, 31, 21, 58], which is integrable via the Bethe ansatz. Here, we are interested in the situation when the exact rsos-constraint is relaxed to the mean ‘spherical constraint’ . In terms of the noise-averaged particle-density, this becomes
where the sums run over all sites of the lattice. Hence, on any site, neither nor the difference can become very large, since the spherical constraint prohibits the condensation of almost all particles onto a very small number of sites. In particular, if one takes a spatially translation-invariant initial condition, then spatial translation-invariance is kept for all times. Because of the constraint (1.7), the average (position-independent) particle-density
is always non-negative. We point out that while the non-averaged density variable has no immediate physical interpretation, the constraint (1.7) guarantees that the measurable disorder-averaged observables takes physically reasonable values.
|KPZ||1||[46, 51, 45, 38]|
The long-time non-equilibrium relaxation behaviour is analysed as follows. In models of interface growth, one usually starts from a flat, horizontal interface with uncorrelated heights [3, 32, 68, 38, 36, 62, 39, 47, 48, 49]. One then studies the average height , the interface width , and the two-time height autocorrelator and auto-response of the height with respect to a change in the height
The scaling forms  used here are those of simple ageing and apply in the long-time limit with being kept fixed. The scaling functions are expected to have the asymptotic behaviour
where is the dynamical exponent. From these relations the exponents and are defined. In table 1, some values of these exponents are collected.555The kpz universality class is realised by the octahedron model . For height correlators and responses, the results of the random sequential (RS) update and of the two-sublattice stochastic dynamics (SCA) update are consistent, confirming the expected universality (, ) . Comparison with the recent result  gives an a posteriori indication of the presently achieved numerical precision. Starting from the Langevin equation (1.4) of the first Arcetri model, formulated in terms of the the slopes and using , an analogous Langevin equation for the heights is found, if only the spherical constraint is now written as . In what follows, we shall call this the Arcetri 1h model. Its relaxational behaviour undergoes (simple) ageing for both and for , in agreement with the expected scaling forms (1.9,1.10). In appendix A, we briefly outline how to find the exponents. Logarithmic sub-scaling exponents  in of the third Arcetri model are discussed in section 4.
The main focus of this work will be on defining (see section 2 for the precise definitions) and analysing the ‘second’ and the ‘third’ Arcetri models. At temperature , we shall see that the simple ageing behaviour of eqs. (1.9,1.10,1.11) does not apply. Rather, we shall find a ‘logarithmic sub-ageing’ behaviour,666Sub-ageing behaviour is defined by the scaling variable , where is the sub-ageing exponent and gives back simple ageing [20, 79]. See [37, Tab 1.2] for a list of experimentally measured values of . A basic rigorous inequality excludes the case (‘super-ageing’) . in the scaling limit where both times , but such that the scaling variable of two-time scaling
is being kept fixed ( is a model-dependent constant). It turns out that several types of logarithmic sub-ageing exist for the Arcetri models, which are characterised by different values of the logarithmic sub-ageing exponent .777For , one is back to simple ageing With the scaling variable (1.12), the asymptotic scaling forms (1.11) often remain applicable and the corresponding exponent values are quoted in tables 1 and 2. Logarithmic sub-ageing arises from the presence of several time-dependent length scales, which differ by factors logarithmic in time, a phenomenon also referred to as multiscaling . If the autocorrelator scaling function decays with faster than a power-law (exponentially or stretched exponentially), the value is quoted. See section 5 for a fuller discussion.
|octa RS kpz||2|||
|octa SCA kpz||2|||
|octa SCA ew||2|||
|octa RS ew||2|||
Analogously, if one considers a system of interacting particles, one usually assumes an initial state of uncorrelated particles (uncorrelated, flat slopes , in the present terminology) with an average particle density , equivalent to a vanishing initial slope [21, 48]. One considers the two-time slope (connected) auto-correlator , which is related to the density-density autocorrelator, and the linear auto-responses , of the slope with respect to a change in the slope or a change in the height, respectively
along with the expected behaviour of simple ageing in the scaling limit. Eq. (1.11) applies again and analogously, one anticipates , for . Considering numerical simulations of the octahedron model, however, it appears that for the slope correlations the two update schemes RS and SCA lead to different values of the autocorrelation exponent – and this for model realisation both in the kpz as well as in the ew universality classes . The first Arcetri model with initially uncorrelated slopes will be called the Arcetri 1U model. It is suggestive to compare the corresponding exponent values with those of the ew universality class. Some values of these exponents are listed in table 2, see appendix A for the outline of the calculations in the Arcetri 1u model. We also include results from the spherical model with a conserved order parameter (‘model B’), at [14, 9]. It also becomes apparent how much less is known about responses of the slope variables than for the height variables.
This work is organised as follows: in section 2, the second and third Arcetri model are carefully defined. Since the first Arcetri model was already studied , we merely outline its treatment in appendix A and quote the results in tables 1 and 2, where the two possible interpretations are taken into account. Section 3 explains the solution of the second and third models. The explicit spherical constraints and a closed form for correlators and responses are derived. In section 4, the asymptotic analysis at temperature and the emergence of the different types of logarithmic sub-ageing in the second and third models is presented. We conclude in section 5 with a detailed presentation of the kinetic phase diagram and the various scales on which different aspects of logarithmic sub-ageing occur. Technical calculations are treated in several appendices. Appendix A contains a short summary of the first model, both for an interface and for a lattice gas. Appendices B and C derive the various distinct sub-ageing scaling forms of correlators and responses, respectively. Several mathematical identities are derived in appendices D and E and some basics of discrete cosine- and sine transformations are collected in appendix F.
2 The second and third Arcetri models
Why are the equations (1.5,1.6) physically unsatisfactory ? In order to understand this, and in consequence the necessity for a better definition of the models, consider for a moment the behaviour of the stationary profiles, as they would follow from eqs. (1.4,1.5,1.6). Let denote the stationary value of the Lagrange multiplier. Then the noise-averaged slope profile of the first Arcetri model (1.4) is oscillatory , with the finite wave-length , as one would have expected. On the other hand, eq. (1.5) would produce a spatially strongly variable stationary slope profile , with a finite length-scale . Finally, eq. (1.6) gives an analogous result for the stationary height profile. This is in apparent contradiction with the expectation of essentially flat profiles, both for the height as well as the slope.
2.2 Definition of the second Arcetri model
How can one formulate a physically sensible ‘spherical model variant’ of the Burgers equation ? Begin with a decomposition of the slope profile , with , into its even and odd parts
For definiteness, we shall formulate the defining equations of motion of the second Arcetri model on a periodic chain of sites. They read
where is the parity-symmetrised and -antisymmetrised white noise , with the moments
Hence one has the (anti-)symmetrised noise correlators
(clearly, the indices are to be taken modulo ). The second Arcetri model will be considered as a variant of the Burgers equation and its associated tasep. Therefore, a natural choice of initial conditions is to admit initially uncorrelated slopes, distributed according to a gaussian, and with the moments
The Lagrange multiplier is determined from the mean spherical constraint on the slopes
which is averaged over both sources of noise present in the model, as indicated by the brackets for the average over and for the average over the initial conditions. We stress that the even and odd parts are treated in a slightly different way. In this way, two essential properties of the Burgers equation, namely (i) the conservation law and (ii) the non-invariance under the parity transformation [10, 4] are kept. The initial conditions (2.6) are natural if one wishes to interpret the slope in terms of the density of a model of interacting particles, with the average density .
Formally, one might also arrive at these equations by introducing a complex velocity into the modification (1.5) of the Burgers equation, with a complex Lagrange multiplier and a complex noise . Separating into real and imaginary parts, this would give
Only if one chooses , one obtains an oscillatory equation for the derivative of the noise-averaged stationary slope, and similarly for . The effect of this formally ‘imaginary’ Lagrange multiplier is included in the equations of motion (2.3).
Conservation laws become explicit by rewriting the complex equations of motion (1.5)
in the form of a continuity equation. Using and its formal complex conjugate , along with and , we have the pair of equations
and identify the densities and of the conserved currents, such that the ‘conserved charges’ and are time-independent, viz. .
2.3 Definition of the third Arcetri model
Analogously, for the third Arcetri model we start from the height profile , decomposed into even and odd parts
and write down the defining equations of motion (on a discrete chain of sites)
with the symmetrised noise (2.5). In this physical context, it appears natural to use initially uncorrelated gaussian slopes
The Lagrange multiplier is found from the mean spherical constraint on the slopes
where is the symmetrised spatial difference. The initial conditions (2.11) are natural for an interpretation of as the height of a growing and fluctuating interface, which is flat on average.
Formally, one might obtain this from the modified kpz equation (1.6) by introducing a complex height , a complex Lagrange multiplier and a complex noise . As before, only if one chooses , the derivative of the stationary height obeys an oscillatory equation . Because of the ‘non-conserved’ noise, there are no obvious conservation laws, for .
All definitions were only made explicit in spatial dimensions. Eventual extensions to are left for future work.
We begin our discussion with the second Arcetri model. The treatment of the third Arcetri model being fairly analogous, we shall simply quote the relevant results in section 3.4.
3.1 Second model: General form
The first step to the solution of eqs. (2.3) proceeds via Fourier-transform, but we must take into account the specific parity of the and . Therefore, we use the representation in terms of discrete cosine- and sine-transforms,
Although we shall use the same notation for both cosine- and sine-transforms, the parity must be taken into account for the inverse transformation. We shall use the short-hands
Later, when taking the continuum limit, it will be enough to simply replace and , and to consider instead of on the chain.
The above equations (3.2) are decoupled by going over to the combinations , which obey the equations
with the solutions
and where the functions are to be found from the initial conditions. Going back to the parity eigenstates, using that and , we have explicitly
and a cosine or sine transformation, respectively, will bring back and . For the chosen initial conditions, we simply have which implies in turn , that is, the interface is always flat on average.
3.2 Second model: spherical constraint
The next step in the solution of the model consists of casting the spherical constraint into an equation for . To do so, the constraint (2.7) is rewritten in Fourier space
Initial conditions must be such that the spherical constraint is respected at , hence
The non-vanishing noise correlators read in Fourier space
such that the constraint can be reexpressed as follows, for this kind of initial condition
The asymptotic analysis of this equation is greatly simplified in the continuum limit, when it takes the form
where the auxiliary functions (3.3) now stand for their continuum versions and .
In what follows, we shall require the following identities, with
3.3 Second model: observables
The observables we are interested in are the two-time correlation and response functions and shall be defined carefully.
For the correlation function, as the order parameter is the local slope , one might expect that should describe the two-time temporal-spatial correlator . However, a physically sensible definition of correlators must obey two symmetry conditions: first, for equal times, the purely spatial correlator