STATISTICAL MECHANICS AND DYNAMICAL COMPLEX SYSTEMS
Statistical Mechanics & Dynamical Complex Systems Laboratory
The Laboratory of Statistical Mechanics and Non-linear Dynamics (STAT-DYN) was founded in 02/2004 as part of the Institute of Physical Chemistry, while from 2014 is part of the Institute of Nanoscience and Nanotechnology. Its research focuses on the fields of :
Our aim is the development of methods and models for understanding the emergence and evolution of mesoscopic and macroscopic spatial and temporal patterns and correlations due to the local interactions between particles at the microscopic level. Such structures include spatiotemporal patterns, aggregates, spiral and stripe formations, helices, fractals, synchronisation phenomena etc. which can be experimentally observed in material science, physics, chemistry and biology. Our studies in particular include research on fractal pattern formation and correlations near the critical point in phase transitions (eg the gas-liquid phase transition) but also research in open systems in constant exchange with the environment. Away from the critical point and in closed, isolated systems, short range correlations and spatiotemporal patterns with well-defined length and time scales are studied (eg. spiral and stripe formations, helices etc.). The study of these structures at the micro-, meso- and macro scales and the interaction between these three levels of description has major technological impact in materials science and physical, chemical and biological processes.
For the study of complex systems in the lab we develop a) statistical methods/tools describing complex morphologies and b) modelling of the dynamics of pattern formation and synchronization phenomena. Statistical methods include thermodynamic approaches, entropic (extensive and non-extensive) approaches, theory of long and short range distributions, and Levi distributions and the theory of random walks. For the study of the mechanisms creating complex patterns, non-linear dynamical systems of hierarchical complexity are used, together with mean-field theories, exact enumeration methods, real space renormalisation theory, theory of stochastic processes and numerical Monte Carlo Methods.
Applications include, among others:
- Studies of surface phenomena and aggregates with fractal morphology,
- Complex networks in reaction-diffusion processes,
- Statistical analysis and modelling of biological tissues and macromolecules,
- Recording and modelling of the complex fractal architecture of the neuron axons spanning the human brain
Typical recent studies are:
1. Complex Networks in Reaction Diffusion Processes
Abstract networks can be used for the description of the dynamics in reactive processes and they can give valuable information about the evolution and the products of each process. The abstract reactive networks are constructed as follows: The phase space of the variables (concentrations in reactive systems) is partitioned into a finite number of segments, which constitute the nodes of the abstract network. Transitions between the nodes are dictated by the dynamics of the reactive process and provide the links between the nodes. These are weighted networks, since each link weight reflects the transition rate between the corresponding states-nodes. With this construction the network properties mirror the dynamics of the underlying process and one can investigate the system properties by studying the corresponding abstract network.
As an example, the abstract network of the Lattice Limit Cycle (LLC) model was analysed and its transition matrix elements were computed via Kinetic (Dynamic) Monte Carlo simulations. For this model it was shown that the degree distribution follows a power law with exponent -1, while the average clustering coefficient c(N) scales with the network size N as with a power law exponent 1.46. The computed exponents classify the LLC abstract reactive network into the scale-free networks. This conclusion corroborates earlier investigations demonstrating the formation of fractal spatial patterns in LLC reactive dynamics due to stochasticity and to the clustering of homologous species. The present construction of abstract networks (based on the partition of the phase space) is generic and can be implemented with appropriate adjustments in many dynamical systems and in time series analysis.
Fig. 2: Representative snapshots of the LLC system from Kinetic Monte Carlo simulations. An initially homogeneous system (t=0 MCS) develops clusters and inhomogeneous distribution of species (t=10000 MCS).
The complexity of the primary structure of human DNA is explored using methods from nonequilibrium statistical mechanics, dynamical systems theory and information theory. A collection of statistical analyses are performed on the DNA data and the results are compared with sequences derived from different stochastic processes. Although detailed balance seems to hold at the level of a binary alphabet it fails when all four basepairs are considered, suggesting spatial asymmetry and irreversibility. Furthermore, the block entropy does not increase linearly with the block size, reflecting the long range nature of the correlations in the human genomic sequences. To probe locally the spatial structure of the chain we study the exit distances from a specific symbol, the distribution of recurrence distances and the Hurst exponent, all of which show power law tails and long range characteristics. These results suggest that human DNA can be viewed as a non-equilibrium structure maintained in its state through interactions with a constantly changing environment.
Fig. 2 : Exemplary Information Transfer Analysis between a sequence and its shift by n symbols in the 4-letter representation. Line with circles depicts the DNA sequence, line with crosses a random sequence and line with diamonds a model-generated sequence
Complexity measures are also used to compare the genomic characteristics of different organisms belonging to distinct classes spanning the evolutionary tree: higher eukaryotes, amoebae, unicellular eukaryotes and bacteria. We demonstrate that the conditional probability matrix for the four-letter and AT-CG alphabet is markedly asymmetric in eukaryotes while it is nearly symmetric in bacterial genomes. Overall, the conditional probability, the fluxes, the block entropy content and the exit distance distributions can be used as markers, discriminating between eukaryotic and prokaryotic DNA, allowing in many cases to discern details related to finer classes. In all cases the reduction from four letters to two mask some important statistical and spatial properties, with the AT-CG alphabet having higher ability of discrimination than the AG-CT one.
3. Recording and modelling of the complex fractal architecture of the neuron axons spanning the human brain
Based on the classical Magnetic Resonance Imaging (MRI) technique it is now possible to obtain high resolution brain images reaching scales as low as ~0.2 mm. Diffusion Tensor MRI tomography (DTI-MRI) allows for capturing the 3D motion of the water diffusion in the brain area by using the anisoptropic movement of water molecules to capture the positions of the axons connecting the neurons. Our studies have demonstrated self-similarity in the form of a hierarchical architecture of the connectivity matrix of the neuron axons network, which is proper to the healthy state of the brain, while its fractal dimension were calculated to df~2.5. Using fractal and multifractal analyses we show that these complex connectivity patterns cannot be a result of simple or even chaotic processes because rare, unexpected configurations are present, indicative of the influence of long range exchange mechanisms. A further challenge is to extend the already existing models to cover the cases of pathological brain architectures (Alzheimer, Parkinson and Schizophrenia), a problem which has not been undertaken thus far.
In parallel with the analysis of the detailed structure of the neuron axons network, we use
Fig. 4. Exemplary chimera state in DTI-MRI imaging hierarchical connectivity networks
numerical simulations to illustrate brain dynamics, using precise network connectivity matrices as recorded in the first part of the project. This hierarchical connectivity in the brain, we consider as a key factor for the development of the intricate spatiotemporal patterns covering multiple scales, necessary for the emergence of cognition. Coupling schemes between a large number of neurons follow three main connectivity patterns: all-to-all (global) coupling, nearest neighbours (or local ) coupling and non-local coupling where each neuron is linked with a finite number of neighbours. The last coupling pattern under special conditions produces the so-called ”chimera states”, where some of the oscillators (neurons) oscillate in phase while others are incoherent. This coexistence of coherent and incoherent states have been associated with partial brain functionality, as in simple brain tasks which engage only a few modules of the brain or in uni-hemispheric sleep. We believe that the presence of chimera states is far more important than proposed for the unihemispheric sleep and is directly related to brain functionality during, at least, simple tasks.
For more information on the STAT-DYN lab see http://limnos.chem.demokritos.gr
Prof. Mandilara Aikaterini
Undergraduate (Bachelor) Students
Previous Students (2015-2016)
Kasimatis Theodoros (MA 2016)
Breki Christina-Marina (BA 2015)
Tsigkri Nefeli-Dimitra (MA 2015)
Flokas Lambros (BA 2015)
Mousa George (BA 2016)
Theodoropoulos Spyros (BA 2016)
Group Photos 2016
Group Photos 2015