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Project submitted for the base funding of
Artem Alikhanyan National Laboratory (ANL)
I.
Principal Investigator: Dr. Davit Sahakyan
TITLE: Statistical physics of disordered systems and its interdisciplinary applications.
Division, group: Theoretical department
DURATION: 36 months
Estimated Project Costs
Estimated total cost of the project (US 45500 $)
Including:
Payments to Individual Participants
36000
Equipment
Materials
Other Direct Costs-books
500
Travel
9000
PROBLEM:
Our main idea is to elaborate advanced methods of statistical physics of disordered systems, and apply
them to the practically important tasks in diverse areas: evolution theory, economics, data mining, energy
storage and conversion. The reason of such wide applications of statistical physics is that it developed
flexible methods for dealing with many-body problems (physical degrees of freedom carrying energy and
information, genes, strategically interacting agents in economics, streams of data).
Within the evolution theory we will focus on investigating problems of virus evolution: extinction threshold,
robustness, the finite population problem, recombination mechanisms. In solving these problems we shall
apply non-perturbative methods of the spin-glass theory, the Hamilton-Jacobi equation formalism, and
quantum field theory. Our solutions will apply to a variety of important practical problems, e.g., predicting
virus diseases, the behavior of HIV virus, etc.
Within the econophysics we shall solve models that have been applied by economists to the analysis of
financial markets (multi-fractal random walk model and Markov switching model). Our hypothesis is that
the crashes in financial markets are connected with specific phase transitions in these models, and, in
principle, could be predicted.
Within the data mining and inference theory we shall develop new, statistical physics-motivated
algorithms for noisy data filtering and for parameter learning in the context of Hidden Markov Models, one
of the main modeling tools in bioinformatics, statistics and information theory. We shall also apply the
methods of statistical physics for analyzing the performance of the existing algorithms for filtration and
parameter learning (e.g., Viterbi and Baum-Welch algorithms). Our basic method will be the theory of
calculating Lyapunov indices (cycle expansion, zeta-functions) and methods developed for random-field
lattice systems.
Within the theory of cooling and energy storage we shall study thermodynamic limits of cooling: how
much one can cool a physical substance given limited energy resources? The primary target of this
research is in designing new approaches for dynamic nuclear polarization in NMR (nuclear magnetic
resonance) physics. This subject of NMR-physics currently undergoes a renaissance due to
implementing dynamic nuclear polarization in liquid-state NMR. Dynamic nuclear polarization presents a
number of physical challenges, which are related to increasing the polarization storage time ( T1
relaxation time) and designing new physical methods for increasing the rate of polarization transfer.
These issues belong to the realm of quantum statistical physics and will be studied within the current
project. The subject promises new applications in MRI (magnetic resonance imaging) based medicine. Next,
we shall determine whether and to which extent one can store useful energy (work) in equilibrium (primarily,
microcanonical) states of a many-body physical system. The results will be applied for designing fast control systems
operating via negative feedback. They will be also useful for developing new principles of energy accumulation and
storage.
II, III. OBJECTIVES and IMPACT:
Virus evolution. We plan to develop an accurate theory for virus evolution that properly accounts for nonperturbative aspects of the phenomenon. This will improve predictions of future virus diseases, and will lead to a
better understanding of the HIV virus and its evolution. In particular, we plan to develop the theory of HIV
virus recombination. Many virologists claim that it is impossible to fight the HIV virus without understanding its
recombination. We plan to give a mathematical theory for the reactions with few molecules, identify the statistical
physics phases in the cell, and try to find some connection with these phases and the cancer phenomenon. We also
plan to identify the statistical physics phases in the cell.
Econophysics. We will solve (preferably exactly) several basic models for the economics of financial
markets. Our exact solutions will improve the accuracy of predictions for future market crashes. In
particular, we plan to find exact solution for the multi-fractal random model.
Statistical physics of data mining and inference. As a result of our research we expect to find new algorithms for
noisy data filtration and parameter learning. These algorithms will advance the fields of bioinformatics and automatic
speech recognition, where it is necessary to have fast inference algorithms that (especially in bioinformatics) are
integrated with the physical content of the problem. The proper theory for several widely applied data mining
algorithms, e.g., the Viterbi algorithm for filtering and expectation-maximization (Baum-Welch) algorithm of parameter
learning, is currently lacking. This prevents the understanding of their limitations and impedes their improvement. Our
results will be relevant in this context.
Cooling and energy storage. We shall develop new methods for dynamic nuclear polarization (spin cooling)
based on the ideas of quantum control and quantum statistical mechanics. These methods can find applications in
MRI (magnetic resonance imaging) that amounts to various methods for structural diagnostics of tissues
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and organs. The main limitation of MRI is a small polarization of the H and/or C spins. This is
precisely the problem addressed by dynamic nuclear polarization. We also expect that our results will lead
to new design principles for energy storage and accumulation.
Participating
Institutions
Task 1: Econophysics
1.1
Kiel University
Solution of Multi-fractal Random walk model.
Kiel University
1.2
Solution of the Markov switching models; determination of its phase structure.
1.3
Solution of continuous branching models and connection with conformal fieldtheoretical models.
1.4
Kiel University
Application of our solutions to analyzing financial data. Comparison of different methods
for predicting financial crashes.
1.5
Non-Hamiltonian statistical mechanics and its possible connection with econophysics.
Research results will be delivered via publications in high-quality scientific journals on statistical physics and economics such as Phys. Rev.
Lett., Physical Review E, JSTAT, The Journal of Financial and Quantitative Analysis, Quantitative Finance
Financial and Quantitative Analysis
Participating
Institutions
Task 2: Virus evolution models
Academia Sinica
2.1
Limits of diffusion theory in population genetics. Fluctuation-dissipation relations in
evolution.
2.2
Collective, non-perturbative phenomena in evolution theory. Evolution for finite
populations.
Rice University
2.3
Recombination phenomena in HIV. Microscopic finite population models of
recombination and the role of lethal mutations. Interference of the mutation
robustness and recombination.
2.4
Extinction threshold versus error threshold for virus populations.
Research results will be delivered via publications in high-quality scientific journals on statistical physics and mathematical/theoretical
biology such as Phys. Rev. Lett., Physical Review E, JSTAT, J. of Theoretical Biology, Mathematical Biosciences, PNAS, PLoS Biology
Task 3: Design properties of the living matter, few-body chemical reactions and
the cell modeling
3.1
Participating
Institutions
Seattle University
Genotype-phenotype mapping and resonance phenomena.
3.2
Seattle University
Stability of ecological communities: new ideas.
3.3
Vienna University
Minimal models of autocatalytic reactions necessary for the life emergence.
Statistical physics phases for the networks of chemical reactions.
3.4
Ecole Polytechnique
Describing chemical reactions with few molecules.
Research results will be delivered via publications in high-quality scientific journals on statistical physics and chemical physics such as
Phys. Rev. Lett., Physical Review E, JSTAT, J. of Chem. Phys., J. of Physical Chemistry, Biophysical Journal
Task 4: Inference and learning problems.
4.1
Theoretical analysis of the Viterbi filtering for discrete-space Hidden Markov Models (HMM)
Participating
Institutions
University of Southern
California (USC)
via lstatistical mechanics. Random-field Ising and Potts models. Applicability of the functional
renormalization group.
4.2
Soliton filtering of continuous-space HMM. Solitons in disordered potential. Non-linear
filtering algorithms. Advantages and shortcomings of realizing filters via non-linear elements.
Generalization of the Kalman’s filter.
4.3
Statistical mechanics of parameter learning for discrete-space HMM. The role of free energy,
University of Southern
California (USC)
temperature and internal energy.
4.4
Statistical mechanics of the Occam’s razor.
Research results will be delivered via publications in journals on statistical physics, artificial intelligence and machine learning such as
Phys. Rev. Lett., Phys. Rev. E, JSTAT, J. of Machine Learning, IEEE-IT. Preliminary results will be reported in conference proceedings.
Task 5: Cooling, polarization transfer and energy storage.
5.1
Thermodynamic limits of cooling: how much one can cool dynamically a quantum system
Participating
Institutions
Stuttgart University
with given energy levels and how much work has to be invested in this maximal cooling.
Feasibility of realizing the optimal cooling schemes.
5.2
How much work can be extracted from a many-body system in a microcanonical
equilibrium state via cyclically acting external fields. The meaning for the non-equivalence
between the canonical and microcanonical ensembles in the context of work extraction.
5.3
Thermodynamics of quantum rotating systems. The thermodynamic meaning of the Coriolis
force.
5.4
Theory of energy storage for biological systems. Energy storage in slowly relaxing Max-Planck Institute for
Biological Cybernetics,
degrees of freedom.
Tuebingen
5.5
Trade-off between efficiency and complexity of molecular energy conversion
processes.
Research results will be delivered via publications in high-quality scientific journals on statistical physics and mathematical/theoretical
biology such as Phys. Rev. Lett., Physical Review E, JSTAT, J. of Theoretical Biology.
IV. The worldwide researches made on the project topic and the
competitiveness of the project
Virus evolution.
Evolution research is necessary for understanding major problems of modern bio-medicine: emergence
of new viruses, HIV and cancer. Within this proposal we shall solve several problems of the evolution
research and apply them to concrete bio-medical problems. Interdisciplinary applications of statistical
physics are important here [1--6], because evolution phenomena have very often essentially collective,
non-perturbative character. They generate subtle structures with non-trivial mathematical description
(“reasonable” approximations popular in mathematical biology, could overestimate the results in
hundreds of times, see [6]). Hence for understanding general features of evolution we need microscopic
models, where many-body effects can be analyzed non-perturbatively. To this end, it is necessary to
employ statistical physics methods. Since 2004 our group actively publishes in this field [7--20].
Among our achievements is the exact solution of the Eigen’s model (an open problem since 1971).
There are several strong groups applying statistical physics to evolution: D. Fisher, M. Deem, L.
Peliti, K. Kaneko, E. Baake, M.Lassig, J. Krug, P. Schuster. We support scientific contacts with most
of them. During the last 5-6 years our group derived more exact results in evolution theory than any
other group. Methods from the spin-glass theory are especially useful here [21--23]. Also here our
group has leading positions [24,25].
Econophysics.
Recent years witnessed applications of statistical physics to the theory of financial markets (groups
by E. Stanley, J.P. Bouchaud, D. Sornette, T.Lux) [26--28]. These groups employed ideas from
scaling theory and spin-glass physics. Our group just started researching applications of statistical
physics to financial markets. In collaboration with T. Lux (a world leader in econophysics) we recently
related the models employed in data analysis of financial markets to spin glasses (Random Energy
Model) and conformal field-theory. These observations will hopefully lead us to exact solutions for
certain well known models of the finance theory (e.g., multi-fractal random walk) [74].
Statistical physics of data mining and inference.
Hidden Markov models (HMMs) is regarded to be one of the most successful statistical modeling tools
that have emerged in the last 40-50 years [29--31]. They consist of two components: hidden states and
observations. The hidden states are unobservable and form a Markov chain. The observations are
independent conditionally on the states and provide the only available information about the state
dynamics. Thus the observed stochastic process is generated by a Markov process observed via a
memory-less noisy channel [29--31]. HMMs are widely employed in various areas of probabilistic
modeling: information theory, signal processing, bioinformatics, mathematical economics, linguistics, etc.
There are two reasons for this. First, HMM present simple and flexible models for a history-dependent
random process. This is in contrast to the Markov process, where the history is irrelevant. Second, HMM
provide the simplest situation, where structured data passes through a noisy channel. The inference
problems of HMM naturally divide into two classes [29--31]: (i) recovering the hidden sequence of states
given the observed sequence (filtering), and (ii) parameter learning, where the transition probabilities of
the hidden Markov chain (and/or conditional probabilities of observations) are unknown and have to be
determined from the observed sequence. There exist concrete methods for each class [29--31] (e.g.,
expectation-maximization algorithm, Baum-Viterbi method, etc). However, not much is known on the
performance of these methods and on their possible limitations. Our premise – already partially
supported via previous results by us and other groups -- is that reformulating these algorithms as
statistical physics models will allow to gain information on their performance and will suggest new
inference algorithms.
Our previous experience in this subject includes developing an efficient method for calculating the
entropy of HMM [32] (which is the first step towards analyzing theoretically inference algorithms), and
statistical mechanics analysis of the Viterbi filtering algorithm [33]. This work already uncovered certain
basic limitations of the Viterbi algorithm. In the field of probabilistic learning we recently carried out a
statistical physics analysis of the so called semi-supervised learning [33]. Earlier, our group worked on
related problems of developing statistical physics-based error-correcting for noisy channels [34--38].
Cooling.
Nuclear Magnetic Resonance (NMR) is applied in many areas of modern science (physics, biology,
chemistry, medicine) [40]. Its application to medicine is best known as MRI (magnetic resonance
imaging) and consists of various methods for structural diagnostics of tissues and organs (proton-NMR,
carbon-13-NMR, chemical shift NMR) [40--42]. The main advantage of MRI compared to computed
tomography and imaging methods based on radioactive tracers (nuclear medicine) is that MRI is able to
record the dynamic state of tissues and organs; in particular, it is able to give real-time resolutions of
various metabolic processes, flow effects (blood circulation) and tissue oxygenation [41,42]. Besides
medical applications, this aspect is crucial for the bio-physical research [42].
Till recently, most applications of MRI techniques are limited to the detection of water, because other
substances are not sufficiently abundant and are not polarized sufficiently strongly. Both these factors
lead to low sensitivity of MRI, which fundamentally originates from the low magnetic energy of nuclear
spins compared with the thermal energy at room temperature [40-42]. Thus the main challenge for the
efficient usage of MRI techniques is to increase dynamically (cooling) the spin polarization of basic NMR
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substances (e.g., C , N , H ) which can be used, in particular, in endogenous medical substances
[42]. The same problem of cooling is of fundamental importance in applications of NMR to quantum
information processors, NMR chemical spectroscopy, etc [40]. The method of dynamic nuclear
polarization (DNP) emerged recently as the main tool for cooling [40,42]. It consists of identifying highly
polarized substances (usually these are electron spins with or without optical pumping; single electron
3
spin polarization is 10 times greater than that of the proton) and transferring this polarization to the
target nuclear spin. The method was known in solid state NMR, but recently it is actively applied also in
the liquid-state NMR that opens up new applications in medicine [42]. Within this proposal we plan to
develop new methods of cooling, and to analyze the principal limits for the existing methods.
A. E. Allahverdyan works on cooling and NMR related areas since 2003. In 2003 he (together with Balian
and Nieuwenhuizen) proposed a new method of spin tomography (state determination), which employed
commutative quantum measurements only and was based on distributing the target spin state between
several spin systems [43]. In 2007 the new method was experimentally realized in the Dortmund NMR
group on a liquid-state NMR system [44]. In 2004 Allahverdyan et al. proposed a new method of
dynamical spin cooling that does not use the conventional polarization resources (e.g., electronic spins),
but instead finds polarization resources in a phononic thermal bath [45]. In 2006 Allahverdyan and
Petrosyan proposed a new mechanism of cooling in two-temperature NMR systems [46]. This
mechanism was based on a new scenario of phase transition for this non-equilibrium system. In 20092010 Allahverdyan (together with Hovhannisyan and Mahler) studied the principal limitations for cooling
arising from quantum mechanics [48,49].
Energy storage and quantum thermodynamics.
The basic purpose of thermodynamics is in studying energy transformation, storage and utilization by
various machines [50--52]. Though in its historical development thermodynamics concentrated on
macroscopic machines, it became nowadays obvious that many important machines are of micro and
nanoscale [52]. Living organisms are integrations of functionally diverse molecular machines, due to
which we can walk, talk, and think. Equally important are the artificial machines constructed on small
scales [52]. A common prediction is that the progress in miniaturization of devices will open the way to
new technologies in various fields. The necessity of understanding the processes of energy storage and
transformation at the nanoscale, with a full account for quantum effects, motivated the development of
the field of quantum thermodynamics; see [50,51] for reviews. This subject studies processes of work
(useful energy) extraction with help of a finite (nanoscale) engine, energy storage in nanoscale batteries,
energy exchange between interacting finite systems, microscopic theories of restitution coefficients, the
energy transfer and storage in rotating systems (encountered in ATP, the basic battery of the biological
energy), etc. Some of these problems will be studied within the present proposal.
A.E. Allahverdyan is one of the founders of quantum thermodynamics. His work with Nieuwenhuizen in
2000 on non-trivial mechanisms of energy storage in the quantum situation initiated this field; see also
[53--65].
V. Personnel Commitments (chart, total number of project participants,
responsibilities of each).
1. Leading researcher Dr. D.B. Sahakyan, the PI of the project.
2. Senior researcher Dr. A. E. Allahverdyan.
3. Researcher Z. Kirakossyan, phd student.
4. Research assistant К. Hovhannisyan, phd student.
The average age of participants is 35, there are 2 participants younger than 35, and nobody is retired.
Work plan:
Task 1: Sahakyan and Kirakossyan (1.1—1.4). Sahakyan and Allahverdyan (1.5).
Task 2: Sahakyan and Kirakossyan.
Task 3: Sahakyan, Allahverdyan and Hovhannisyan.
Task 4: Allahverdyan (4.1—4.4). Sahakyan (4.4)
Task 5: Allahverdyan and Hovhannisyan (5.1—5.5). Sahakyan (5.4).
Equipment
200 $
3 Cartrages for the printer
Total
Подробное описание заказываемого оборудования, указание сайта фирмы.
Materials
Paper, pens
150 $
Direct cost description
Cost (US $)
Other Direct Costs
Travel costs (US $)
CIS travel
International travel
9000
Total
9000
VI Technical Approach and Methodology
Task 1: Econophysics is a field, where the problems of economics are studied using methods of
theoretical (in particular, statistical) physics. Statistical physics is applicable, because there are many
degrees of freedom (agents, goods) and quantities that have certain features of energy (money,
value). The main postulate of statistical physics is the ergodic/mixing hypothesis (maximum entropy
principle). Its analogue in economics is the concept of self-regulating market. One of the main tools of
statistical physics is the scaling relations, which were successfully applied to economics in [27].
Mandelbrot and co-authors suggested to apply multi-fractal models for describing stock fluctuations in
financial markets [28]. This is more realistic than modeling them via Brownian motion. Further
advances in this field relate to Markov-switching multi-fractal models (MSM), which recover the timetranslation symmetry lacking in the previous models. Nowadays MSM is a very popular tool for
analyzing market data [74].
According to our preliminary results, there is an exact mapping between multi-scaling models and the
Random Energy Model [21--23] (one of the paradigmatic models of the spin-glass theory). This
mapping can be used for improving the accuracy of econometric calculations, identifying the market’s
phase structure, and predicting financial crashes. Our idea is to identify those crashes by looking to
the high frequency fluctuations of the market. In the statistical physics language this corresponds to
the spin-glass phase transition. When mapping the concepts of economics to statistical
thermodynamics, we realized that the main difference is that the first law of thermodynamics is lacking
in economics. This is why we will try to use a weaker form of statistical mechanics (non-Hamiltonian
statistical mechanics) for modeling economics.
Task 2.1: The diffusion equation is certainly the main working mathematical method of population
genetics [94,95]. Within this task we shall understand whether this method applies to the strong selection
situation, which is typical for viruses and bacteria. According to our preliminary results it does not apply
and should be replaced by the Hamilton-Jacobi approach [12].
Task 2.2: The finite population problem is one of the central topics in the evolution research. Recently
there have been several serious attempts of solving this problem. They however contradict each other
[96,97]. Ref. [97] widely used the solitary wave concept, which is borrowed from the infinite population
evolution research. As we have carefully checked in our article [16], the concept of solitary wave is
completely wrong for infinite populations. We plan to perform some numeric and theoretical analysis to
clarify the applicability of this concept to finite populations.
Task 2.3: Here we should understand the role of robustness in evolution systems and its relation to
recombination. There were many theoretical speculations about possibility of a selection via flatness
instead of selection via fitness. However, such a situation was experimentally observed only recently
[89]. Interestingly, the robustness can increase the role of recombination [90]; see also the recent review
about recombination phenomenon [91]. We will study the interference of recombination and robustness
via an analytical theory. This will lead to understanding dynamic aspects of robustness.
Within this task we shall also develop a simplified theory of recombination. To this end we will consider a
quasi-linkage equilibrium situation, suggested by Kimura, and apply to it fluctuation-dissipation relations.
We observe that different quantities in evolution have either mean-field or fluctuating behavior. We will
use this observation to simplify complicated mathematical machinery of recombination theory by ideas
and methods of statistical mechanics.
Task 2.4: There are two views among the virologists about extinction of virus populations at high
mutation rate. It could be explained using either the concept of error threshold, or the extinction threshold
[92,93]. We shall understand how to define the correct extinction threshold in the case of lethality. This
issue is important for both virology and the origin of life research.
Task 3.1: Understanding of the genotype-phenotype mapping is an important subject of bio-medical
research [75,76]. There is much room here for new ideas and approached from statistical physics, e.g.,
Ref. [75] suggested that the phenotype properties were shaped together with mutational robustness
during the evolution. We shall study the genotype-phenotype mapping from the information theoretical
viewpoint, employing our previous experience in relating statistical physics to information theory [35--39].
We shall attempt to define a design temperature for genotype-phenotype mapping, and try to understand
the role of RNA in this mapping.
Task 3.2: This problem arose when ecological modelers noted that substantially random interaction
between various species will never support the population stability observed in nature [77,78]. Hence one
should look for some design of inter-species interaction, which came out during evolution. We shall study
a modified version of the multi-species interaction model [78] from the information -theoretical viewpoint.
In particular, we shall attempt to describe how information is transferred from the couplings of multispecies interaction to the frequencies of different species.
Task 3.3: Ref. [79] proposed to decribe the living matter as a set of auto-catalyzing reactions. This idea
was elaborated in [80]. Later on Ref. [81] suggested modeling the cell trough a limit cycle of the proteinprotein interaction network; see [82] for a review. There is a substantial room for new ideas here, e.g.,
because the microscopic mechanism of the cell differentiation [83] is not captured by the existing models
[82]. Our plan is to analyze the minimal version of autocatalytic reactions necessary for the life modeling.
We will also analyze all possible phases for the autocatalytic reactions [84--86] and select the most
relevant one for the emergence of life. We expect that different aspects of the cell dynamics -metabolism, information processing, control and the interaction with other cells -- are reflected in the
internal hidden state of the cell. We also expect that our ideas will be useful for understanding the
purposeful behavior, the main characteristic feature of the living matter.
Task 3.4: The number of molecules participating in the cell dynamics is sometimes rather small. This
makes inapplicable the standard methods of chemical physics. We shall apply to this situation the
Hamilton-Jacobi approach [87], which was recently developed in our work [12]. We are interested mainly
by chemical master equation with two connected chains of equations [88].
Task 4.1: We shall employ methods from the theory of disordered lattice models (statistical
mechanics). This includes functional renormalization group [69], embedded Markov technique [66],
etc.
Task 4.2: The specific point of this problem is that one ought to study solitons moving i n a disordered
potential [68]. Hence we shall most probably have to work out a generalization of standard soliton
theory methods [67] (e.g., inverse scattering method) to this situation.
Task 4.3: We intend to investigate this problem via the zeta-function method [66] for Hidden Markov
Models [32].
Task 4.4: The problem will be approached via the disorder-dominated renormalization group [69] and
the Bayesian field theory [72,73].
Tasks 5.1 – 5.2: For understanding this problem we need to solve a (typically high-dimensional)
optimization problem in the joint space of physical states (density transformations) and dynamic
transformations (unitary operators). Work is currently in progress to develop effective optimization
methods for this solving such problems.
Task 5.3: We need to understand foundations of statistical mechanics as seen from a non-intertial
frame.
Task 5.4: A detailed understanding of the meta-stability theory for Markov chains is needed [70]. We
also expect to use the majorization theory results.
Task 5.5: We shall approach this problem from the stands of modern computational complexity [71].
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4.
E. Baake, W. Gabriel, Ann. Review Comp. Phys. 7,203 (2000).
5.
B. Drossel, Advances in Physics 50, 209 (2001).
6. D.B. Saakian, C.K. Hu, Eigen model as a quantum spin chain: exact dynamics, Phys. Rev. E, 69, 021913 (2004).
7. D.B.Saakian and C.K. Hu, Solvable biological evolution model with a parallel mutation-selection scheme, Phys. Rev. E, 69,
046121 (2004).
8. D.B. Saakian, C.K. Hu and H. Khachatryan, Solvable biological evolution models with general fitness functions and multiple
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73, 041913 (2006).
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(USA) 103, 4935 (2006).
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Smooth Fitness Landscapes,
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D.B. Saakian, O. Rozanova, A. Akmetzhanov, Exactly solvable dynamics of the
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theoretical aspects, Phys. Rev. E 80 041903 (2009).
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evolutions due to the finite generation-time effect.
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symmetric or random fitness landscapes.
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with finite population.
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26. P. Jackel, Monte-Carlo Methods in Finance (J. Wiley and Sons, Sussex, 2002)
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