Gradient pattern analysis of structural dynamics: application to molecular system relaxation

Gradient pattern analysis of structural dynamics: application to molecular system relaxation

Reinaldo R. Rosa^I; Marcia R. Campos^I; Fernando M. Ramos^I; Nandamudi L. Vijaykumar^I; Susumu Fujiwara^II; Tetsuya Sato^III

^I Nucleus for Simulation and Analysis of Complex Systems (NUSASC) Laboratory for Computing and Applied Mathematics (LAC) National Institute for Space Research (INPE), 12245-970 S.J. dos Campos, SP, Brazil

^IIDepartment of Polymer Science and Engineering, Faculty of Textile Science Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606-8585, Japan

^IIIJapan Marine Science Technology Center Earth Simulator Center, 3173-25 Showa-machi, Kanazawa-ku, Yokohama 236-0001, Japan

ABSTRACT

This paper describes an innovative technique, the gradient pattern analysis (GPA), for analysing spatially extended dynamics. The measures obtained from GPA are based on the spatio-temporal correlations between large and small amplitude fluctuations of the structure represented as a dynamical gradient pattern. By means of four gradient moments it is possible to quantify the relative fluctuations and scaling coherence at a dynamical numerical lattice and this is a set of proper measures of the pattern complexity and equilibrium. The GPA technique is applied for the first time in 3D-simulated molecular chains with the objective of characterizing small symmetry breaking, amplitude and phase disorder due to spatio-temporal fluctuations driven by the spatially extended dynamics of a relaxation regime.

I Introduction

Spatiotemporal systems driven away from thermodynamic equilibrium can form complex structures when forced beyond a critical threshold. The spatiotemporal dynamics of such nonequelibrium structures have been the subject of numerous experimental and theoretical investigations, notably in extended neutral and ionized fluid flows, lasers and optical electronics, oscillated granular layers, molecular clusters and percolative systems [1-5]. There are systems in which the instantaneous patterns are disordered, but they retain sufficient phase coherence that the time-averaged images reveal spatially regular and quasi-periodic structures. In the more general case the global structures can exhibit complex spatio-temporal dependencies reflecting many structural nonlinear properties as spatio-temporal fluctuations, symmetry breaking under energy dissipation, scaling coherence and structural entropy.

Thus, quantitative characterization of spatio-temporal patterns is clearly essential to the understanding of spatio-temporal phenomena. An important question in this problem concerns the long-term evolution of the pattern properties. Usually, the classical measures of complex extended variability do not take into account the directional information contained in a vectorial field: the main source of spatio-temporal variability. Moreover, since spatio-temporal information is even more accessible through high resolution digitized images, the need for sensitive techniques working in the real space is evident [6].

The gradient pattern analysis (GPA) is an innovative technique, which characterizes the formation and evolution of extended patterns based on the spatio-temporal correlations between large and small amplitude fluctuations of the structure represented as a gradient field. Here we report, mainly, the performance of this new technique in characterizing nonlinear emergence of ordered structures as from random initial condition, a macroscopic signature called spatio-temporal relaxation (STR). Usually, the STR is a complex spatio-temporal regime described by means of the correlation among many localized dynamics. Here we analyze the STR regime observed from a simulated short chain-molecule system [4]. The chain molecules model consists of a sequence of CH₂ groups, which are treated as united oscillons. The mass of each molecular oscillon is 14 g/mol and they interact via bonded potentials (bond-stretching, bond-bending, and torsional potentials) and a non-bondend Lennard Jones potential. The system, at first, is randomly distributed configuration of 100 short chain molecules, each of which consists of 20 oscillons, at 700-300 K bath temperature. The local fluctuation of this molecular system is given by with D_i,j = where I is the moment of inertia of the system, k_i,j is the local term of Lennard-Jones potential and D_i,j is the term related to the local fluctuations of the rigidity (R) of the chains. More details on the system simulation is given by Fujiwara et al. [4,7].

In Figure 1 shows the chain of 3D-configurations at various times at T=400 K. At the early time, the spatiotemporal configuration is disordered. As time elapses (mainly after t ~ 10²ps), growth of local orientationally-ordered structures is observed in several positions. At last (t ~ 10³ ps), several clusters coalesce into a large structure and a highly ordered pattern is formed. This 3D STR regime is also well characterized as a similar stepwise behaviour taking the 2D cross section from the middle of the system (See Fig.2). Therefore, it becomes interesting to characterize such extended dynamics by means of measures that produce scalar moments from two-dimensional patterns (operations ² Â), consequently obtaining an easily temporal representation of their corresponding dissipative correlations working under far-from-equilibrium conditions.

II The gradient pattern analysis formalism

II.1 The gradient moments

A spatially square extended pattern in two dimensions (x,y) is represented by the square matrix of amplitudes = {M(1,1), ... , M(i,j), ...,M()} \mid M Î Â, essentially a square lattice, , if the two dimensions, x and y, are discretized into × pixels, with i = 1,..., and j = 1,...,. Thus, a dynamical sequence of N lattices, ₀, ₁, ... , _N, is related to the temporal evolution of a visualized amplitude envelope _x,y,t. Usually, each amplitude intensity (i,j) reflects a local measurement of spatially distributed energy, as for example scalar relative velocity, concentration rate, etc. The spatial fluctuation of the global pattern (x,y), at a given instant t, can be characterized by its gradient vector field _t = Ñ[ (x,y)]_t. The local spatial fluctuations, between a pair of pixels, of the global pattern is characterized by its gradient vector at corresponding mesh-points in the two-dimensional space. In this representation, the relative values between pixels (Dº |(i,j)-(i+n,j+n)|) are dynamically relevant, rather than the pixels absolute values. Note that, in a gradient field such relative values, D, can be characterized by each local vector norm and its orientation.

In view of GPA formalism a gradient vector field

_t = Ñ[ (x,y)]_t, is composed by V vectors r where a vector r_i,j is represented, besides its location (i,j) in the lattice, by its norm (r_i,j) and phase (f_i,j), so that associated to each position in the lattice we have a respective vector (r_i,j = (r_i,j, f_i,j)). Thus, a given scalar field of absolute values can be represented by a gradient field for the local amplitude fluctuations, and this gradient pattern can be represented by a pair of matrices, one for the norms and another for the phases. A natural subsequent representation is by means of a complex matrix, where each element corresponds to the respective complex number z_i,j representing each vector from the gradient pattern. Thus, a given matricial scalar field can be represented as a composition of four gradient moments: first order, g₁, is the global representation of the vectors distribution; second order, g₂, is the global representation of the norms; third order, g₃, is the global representation of the phases; and fourth order, g₄, is the global complex representation of the gradient pattern. Considering the sets of {r_i,j} and {f_i,j} as discrete compact groups, spatially distributed in a lattice at instant t, the gradient moments are equivalent to Haar-like measures, h, which has the basic property of being, at least, rotational invariant:

From the definition of the gradient moments g_z, with z = 1,...,4, it is possible to represent the gradient field _t = Ñ[(x,y)]_t as a set of four gradient moments: _t º (g₁, g₂, g₃, g₄)_t (see Figure 3).

II.2 Computational operations to extract the gradient moments

II.2.1. The first gradient moment

As we are interested in nonlinear spatio-temporal structures we have introduced a computational operator to estimate the gradient moment g₁ based on the asymmetries among the vectors of the gradient field of the scalar fluctuations. A global gradient asymmetry measurement, can be performed by means of the asymmetric amplitude fragmentation (AAF) operator[8,9]. This computational operator measures the symmetry breaking of a given dynamical pattern and has been used in many applications [9-14]. From the Ñ () the symmetric pairs of vectors (i.e., the pairs of vectors that have the same modulus but opposite directions) are removed, obtaining the asymmetric field of vectors Ñ_A(). The measurement of asymmetric spatial fragmentation (usually called, the asymmetric amplitude fragmentation F_A) is defined as:

where V_A is the number of asymmetric vectors and C is the number of correlation bars generated by a Delaunay triangulation having the middle point of the asymmetric vectors as vertices. The Delaunay triangulation T_D(C,V_A) is a fractional field with dimension less than two - the lattice dimension [8]. When there is no asymmetric correlation in the pattern, the total number of asymmetric vectors is zero, and then, by definition, is null. For a given × lattice size, while a random and totally disordered pattern has the highest value, a totally ordered and locally symmetric pattern (e.g. an extended Gaussian or Besselian envelope) has equals to zero; and complex patterns composed by locally asymmetric structures has specific nonzero values of .

II.2.2. The fourth gradient moment

In this paper we are not interested in measurements of the second and third gradient moments because still there are no formal computational operators to calculate them in the literature. However, from a generalization of the concept of degeneracy W, given by the multinomial coefficient formula, and normally used to deduce the expression of Shannon's entropy of positive scalar fields, such as numerical lattices, Ramos et al.[9] have introduced a computational operator to estimate the modulus and phase related to the complex form of the fourth gradient moment, g₄. Considering the gradient Ñ() defined above, W(z_1,1,...,z) may be generalized as follows:

where z = å_i,j z_i,j. Using Stirling's approximation, we immediately have

We can easily verify that this complex entropic form (CEF), S_zº g₄ = |g₄|, is invariant under rotation and scaling of the gradient field. Extensive applications of CEF in chaotic coupled map lattices [9] and on solutions of the Swift-Hohenberg equation [10] have shown that: (i) the |g₄| measurement is very sensitive to the spatio-temporal relaxation processes preserving the information on nonlinear local amplitude fluctuations and (ii) the F_g4 measurement characterizes, by means of phase disorder, the transitions from amplitude to phase dynamics. In the presence of spatio-temporal nonlinear fluctuations, such measures from a gradient field on a lattice is more robust than derivative measures and spatial correlation lengths. Particularly, several calculations on random patterns have shown that, in particular, g₁ and F_g4 are much more sensitive and precise in characterizing asymmetric structures than the correlation length measures [8,9].

II.2.3. Gradient scale invariance

Whenever the statistics for the configurations of a complex gradient pattern (composed by V_A asymmetric vectors distributed in a global 2D-scale R×R) has the property that, by scaling the normalized gradient moments _z, one can make the analysis for a smaller sub-pattern (local arbitrary scale r - See Fig. 1) exactly matching that of a larger scale, then the gradient statistics is said to be scale invariant.

At a possible gradient critical point, the two-point correlation function for local macroscopic observable (here, the _z gradient moments), G(r,R) = á _z(r) _z(R) ñ, tall off as some power of |r-R|, that is á_z(r) _z(R) ñ ~ |r-R|-, where the statistical average, indicated by the brackets, is over the fluctuation of norm, phase and symmetry of the gradient field Ñ[ (x,y)]_t. Note that, also a time-dependent correlation function can be defined.

This formalism implies that the fundamental assumption of renormalization group theory can be considered into the context of the gradient pattern analysis. A direct consequence is that in a Landau-Ginzburg approach the gradient moments

_z will describe local disorder that are averages over larger and larger scales. As the scales become much larger than a characteristic correlation length, the averages over different domains become statistically independent and the concept of fixed point can be considered into the GPA formalism.

III Results and interpretation

Next we show the characterization of extended relaxation regimes by means of t(, |g₄| and F_g4). We applied the CEF and AAF operators on the spatio-temporal series, partially illustrated in Fig. 1. For a proper application of the operators [8,9], the relaxation process is described by a sequence of many lattices (here, 20 frames), each of which consists of a 64×64 matrix of real values, representing the sectional state of the system at a given instant. In Figure 4 is shows, for the time-step 15, the asymmetric gradient field and its correpondent triangulation field, from where,by means of CEF and AAF computational operators, it is possible to determine the gradient moments g₁, |g₄| and F_g4.

The global relaxation process may or not be dominated by the amplitude dynamics and also by the influence of the boundary on the local pattern equilibrium. For local abnormal relaxation there are many small sub-relaxation regimes which affect the global pattern equilibrium (characterized when the time derivative of the gradient moments (g_z) is null). An useful description of the relaxation process is given by: (i) the temporal variation of |g₄| shows the influence of the amplitude dynamics on the STR evolution; (ii) the equilibrium pattern (local and global) is characterized by the dynamics in the plane g₁×F_g4.

The short chain-molecule system quickly evolves from a totally disordered state to a state exhibiting several domains of oriented structures with quasi-regular hexagonal boundary. The molecular oscillons interact slowly until only one domain with the same orientation and typical aspect ratio is resulted. Figures ^5a-^b illustrate the results of gradient pattern analysis applied to these data. In ^{Figures 5a}, we plot the values of |g₄| for each frame in the series. The molecular x-y layer starts with disordered oscillatory amplitude moving to quasi-ordered oscillatory amplitudes. During the first integration steps, there is a variation in the maximum amplitude of the oscillons, while the system nucleates to cellular structures. This situation is accurately characterized by the time evolution of the gradient moment |g₄|, as shown in ^Fig.5a. After some random accomodation (into the interval 1 ps < t < 5×10² ps) of the system, due to the spatio-temporal potential arrangement, the effective relaxation starts and reaches the relaxation straight line showed in ^{Fig. 5a}, characterized by |g₄| = 0.945±0.009. ^{Figure 5b} shows that the global pattern becames approximately stable around a fixed point in the g₁×F_g4 space. From previous application of GPA on chaotic coupled map lattices and amplitude equations [9,10] it means that for the molecular oscillons analysed here there is a meta-stable amplitude dynamics represented in the g₁× F_g4 space (^{Fig. 5b}). From the temporal behaviour of |g₄| we found the transition from the dominant norm regime to the dominant phase regime well characterized in the meta-stable g₁×F_g4 space dynamics. During the relaxation the phase correlations become stronger and the averaged equilibrium pattern shows, approximately, a characteristic local geometric scale invariance responsable for the global pattern symmetry.

IV Concluding remarks

The spatio-temporal relaxation into a stable state takes place when the system is instantaneously changed into a nonequilibrium state, and this universal regime is often observed in many physically different systems [e.g., 1-15]. Therefore when a extended system is driven far from equilibrium, it will evolve to a minimum energy state in a stepwise relaxation and, generically, this self-organized behaviour is independent of the types of oscillons and the nature of their interactions.

In short, the gradient pattern analysis specifies quantitatively the relative fluctuations and scaling coherence at a dynamical numerical lattice and this is a proper measure of the global pattern variability due to small fluctuations in the lattice gradient field. The gradient pattern analysis is still under detailed investigation in order to make more precise the concept of characterizing spatio-temporal disorder (nonlinear pattern equilibria, topological intermittency, entropy and chaos) in extended systems. In this sense such gradient moments and their characteristic two-point correlation functions can be considered as a plus to other spatio-temporal measures (e.g. disorder functions [15]). A natural way of extending the GPA application would be analysing magnetization patterns on a two-dimensional Ising lattice characterizing the rate of spatial variation (the gradient) of the magnetization. This new phenomenological approach is also in progress and will be communicated later.

Acknowledgments

Thank are due to A. Mattedi, E.L. Rempel, L. Vinhas and R.C. Brito for some computer work and to J.Pontes, P.M.C de Oliveira, C. Tsallis, D. Stauffer and H.L. Swinney for important discussion on GPA and to P.G. Pascutti on molecular dynamics. This work was partially supported by the Brazilian Agencies FAPESP (grants 97/13374-1 and 98/03104-0) and CNPq.

Received on 30 April, 2003

• [1] D. Walgraef, Spatio-Temporal Pattern Formation, Springer-Verlag, N.Y., 1997.
• [2] M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys, 65, 851 (1993).
• [3] P.B. Umbanhowar, F. Melo, H.L. Swinney, Nature 382(6594), 793 (1996).
• [4] S. Fujiwara and T. Sato, Phys. Rev. Lett. 80, 991 (1998).
• [5] D. Stauffer and A. Aharony, Introduction to Percolation Theory, Taylor & Francis Ltd, PA, USA, 1994.
• [6] K.R. Mecke and D. Stoyan (Editors), Statistical Physics and Spatial Statistics, Lecture Notes in Physics 554, Springer-Verlag, N.Y., 2000.
• [7] S. Fujiwara and T. Sato, Molecular Simulation 21, 271 (1999).
• [8] R.R. Rosa, A.S. Sharma, J.A. Valdivia, Int. J. Mod. Phys.C, 10(1), 147 (1999).
• [9] F.M. Ramos, R.R. Rosa, C. Rodrigues Neto, A. Zanandrea, Physica A, 283, 171 (2000).
• [10] R.R. Rosa, J. Pontes, C.I. Christov, F.M. Ramos, C. Rodrigues Neto, E.L. Rempel, D. Walgraef, Physica A 283, 156 (2000).
• [11] R.R. Rosa, A.S. Sharma, J.A. Valdivia, Physica A 257, 509 (1998).
• [12] C. Rodrigues-Neto, R.R. Rosa, F.M. Ramos, Int. J. Mod. Phys.C 12(8), 1261 (2001).
• [13] A.T. Assireu, R.R. Rosa, N.L. Vijaykumar, J.A. Lorenzzetti, E.L. Rempel, F.M. Ramos, L.D.A. Sa, M.J.A. Bolzan, A. Zanandrea, Physica D 168, 397 (2002).
• [14] A. Ferreira da Silva et al., Sol. Stat. Comm. 113(12), 703 (2000).
• [15] G.H. Gunaratne, R.E. Jones, Q. Ouyang, and H.L. Swinney, Phys. Rev. Lett. 75, 3281 (1995).

Publication Dates

• Publication in this collection
  11 Nov 2003
• Date of issue
  Sept 2003

History

• Received
  30 Apr 2003
• Accepted
  30 Apr 2003