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The recent inflow of empirical data about the collective behaviour of strongly correlated biological systems has brought field theory and the renormalization group into the biophysical arena. Experiments on bird flocks and insect swarms show that social forces act on the particles' velocity through the generator of its rotations, namely the spin, indicating that mode-coupling field theories are necessary to reproduce the correct dynamical behaviour. Unfortunately, a theory for three coupled fields - density, velocity and spin - has a prohibitive degree of intricacy. A simplifying path consists in getting rid of density fluctuations by studying incompressible systems. This requires imposing a solenoidal constraint on the primary field, an unsolved problem even for equilibrium mode-coupling theories. Here, we perform an equilibrium dynamic renormalization group analysis of a mode-coupling field theory subject to a solenoidal constraint; using the classification of Halperin and Hohenberg, we can dub this case as a solenoidal Model G. We demonstrate that the constraint produces a new vertex that mixes static and dynamical coupling constants, and that this vertex is essential to grant the closure of the renormalization group structure and the consistency of dynamics with statics. Interestingly, although the solenoidal constraint leads to a modification of the static universality class, we find that it does not change the dynamical universality class, a result that seems to represent an exception to the general rule that dynamical universality classes are narrower than static ones. Our results constitute a solid stepping stone in the admittedly large chasm towards developing an off-equilibrium mode-coupling theory of biological groups., Instituto de Física de Líquidos y Sistemas Biológicos

The astonishing efficiency of the auditory organ of mammals is particularlydue to the generic properties of the coupled critical oscillators which make upthe system. This thesis presents a study of the generic critical properties ofspatially extended systems of coupled stochastic oscillators, operating in theproximity of a uniform oscillatory instability or Hopf bifurcation. In thiscontext, this bifurcation constitutes an out of equilibrium critical pointwith universal features, which are canonically described by the complexGinzburg-Landau equation in the presence of noise. The formulation of theproblem in terms of a non-Hamiltonian dynamical statistical field theoryallows us to study the critical behavior of the system by using perturbativerenormalization group techniques.In a particular case, an exact analogy with the O(2) dynamical model allowsus to write a generalized fluctuation-dissipation relation and to deduce thecritical behavior directly from previous studies. In the generalcase, we establish the structure of the renormalization group of the theory ina 4-epsilon dimensional space, using adapted Wilson and Callan-Symanzikschemes. The presence of a characteristic frequency in the system - thefrequency of the spontaneous oscillations at the transition - imposes toperform a scale-dependant frame transformation during the renormalizationprocedure. We perform two-loop order calculations in perturbation theory, andshow that the universality class of the model is described, in a suitedoscillating frame, by the fixed point of the dissipative O(2)dynamics. Then, while the dynamics is highly out of equilibrium and breaks thedetailed-balance relations, a generalized fluctuation-dissipation relation isasymptotically recovered at the transition. This relation imposes strongconstraints on the main experimental observables: the two-point correlationfunction and the linear response function to an external sinusoidal stimulus.; Les etonnantes performances de l'organe auditif des mammiferes sontnotamment dues aux proprietes generiques des oscillateurs critiquescouples qui constituent le systeme. Cette these presente une etudedes proprietes critiques generiques dessystemes spatialement etendus d'oscillateurs stochastiques couples,operant dans le voisinage d'une instabilite oscillante homogene oubifurcation de Hopf. Dans ce contexte, cette bifurcation constitue unpoint critique dynamique hors equilibre, exhibant des proprietesuniverselles qui sont canoniquement decrites par l'equationGinzburg-Landau complexe en presence de bruit. La formulation du problemeen termes d'une theorie statistique dynamique des champs non hamiltoniennenous permet d'etudier le comportement critique du systeme a l'aide destechniques de la renormalisation dynamique perturbative.Dans un cas particulier, une analogie exacte avec le modele O(2) dynamiquenous permet d'ecrire une relation generalisee de la relationfluctuation-dissipation et de deduire le comportement critique du systemedirectement a partir des etudes anterieures. Dans le cas general,nous etablissons la structure du groupe de renormalisation de la theoriedans un espace de dimension4-epsilon, en lui adaptant les schemas de renormalisation de Wilson etde Callan-Symanzik. La presence d'une frequence caracteristique dans lesysteme - la frequence des oscillations spontanees a la transition -impose d'associer aux transformations de renormalisation un changement dereferentiel oscillant dependant de l'echelle. Nous effectuons lecalcul a l'ordre de deux boucles en theorie des perturbations, et montronsque la classe d'universalite du modele est decrite par le point fixe dumodele dynamique dissipatifO(2) dans un referentiel oscillant bien choisi. Ainsi, bien que ladynamique soit hautement hors equilibre et brise les relations de bilandetaille, une relation fluctuation-dissipation generalisee estasymptotiquement restauree a la transition. Cette relation prevoitl'existence de fortes contraintes sur les principales observablesexperimentales : la fonction de correlation a deux points et la fonctionde reponse lineaire a un stimulus sinusoidal.

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2003, Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2003, Biyolojik sistemlerde protein üretimi, yani gen ifadesi, birebir olmamakta ve genlerin birbirleriyle olan etkileşimleri sonucu ortaya çıkmaktadır. Bu etkileşimler için, seçilim baskısı olmaksızın mutasyon geçiren basit bir modelde, Monte Carlo simülasyonlarıyla, gen etkileşimlerinin ölçekten bağımsız bir şebeke oluşturduğu ve bir gen tarafından baskılanan gen sayısı ( O z ) dağılımının ( ) O O z τ − ( O τ =0.47±0.01) gibi bir kuvvet yasasına tabi olduğu ve bu şebekenin bir küçük dünya modeline uygun olarak kümelenme katsayısının C ≈ 0.5 olduğu bulundu. Gen etkileşimi şebekesinin bir kinetik Ising modeli biçiminde betimlenmesi durumunda, sistemin tepkimelerinin zaman içinde ölçekten bağımsız bir sönümlenme zamanı dağılımına sahip olabileceği görüldü. Bu kendiliğinden kritik sistemin dinamik davranışını inceleyebilmek için, dinamik renormalizasyon grubu dönüşümleri rastgele Γ - komşuluklu spin sistemlerine genelleştirildi., We have studied the regulatory network of gene expression during protein synthesis in biological systems. We have intoduced a simple random bit-string model to represent a chromosome sequence and have only considered the inhibition interactions between the genes. As the result of our Monte Carlo simulations, we have found that this simple model gives rise to a network of gene expression which is of the small-world type, with a clustering coefficient C ≈ 0.5, and is scale-invariant with the distribution of out-going connectivities obeying n(zout) ∼ (zout)-τ . The exponent τ is found to be τ=0.47±0.01. This result shows that the system is complex and can respond on any scale to the sitimuli coming from the environment. This also shows that the system has a selforganized critical behavior. In order to test ideas regarding the distribution of relaxation times on such a network, we have generalized the dynamical renormalization-group calculations to networks with an arbitrary number of nearest neighbors. This will enable us to compute the dynamical exponent on networks with random connections., Yüksek Lisans, M.Sc.