Physics of Nonequilibrium and Disorder

An important subject of modern statistical physics and a focus of our work are systems away from normal thermodynamic equilibrium, such as growth phenomena and `frozen' disordered materials. These systems are ubiquitous in nature, and they are a challenge for the theorist since traditional methods often fail. Our past work in this area has focused on nonperturbative aspects of surface growth and turbulence. Nonequilibrium statistical mechanics is also crucial for a quantitative understanding of biological evolution and enters much of our current work. Recently, we have proved a fluctuation theorem for stochastic evolutionary processes of biological systems.

Selected physics publications

Dynamical anomalies and intermittency in Burgers turbulence

M. Lässig, Phys. Rev. Lett. 84, 2618, (2000)

We analyze the field theory of fully developed Burgers turbulence. Its key elements are shock fields, which characterize the singularity statistics of the velocity field. The shock fields enter an operator product expansion describing intermittency. The latter is found to be constrained by dynamical anomalies expressing finite dissipation in the inviscid limit. The link between dynamical anomalies and intermittency is argued to be important in a wider context of turbulence.

Quantized scaling of growing surfaces

M. Lässig, Phys Rev. Lett. 80, 2366 (1998)

The Kardar-Parisi-Zhang universality class of stochastic surface growth is studied by exact field-theoretic methods. From previous numerical results, a few qualitative assumptions are inferred. In particular, height correlations should satisfy an operator product expansion and, unlike the correlations in a turbulent fluid, exhibit no multiscaling. These properties impose a quantization condition on the roughness exponent χ and the dynamic exponent z. Hence the exact values χ = 2/5, z = 8/5 for two-dimensional and χ = 2/7, z = 12/7 for three-dimensional surfaces are derived.

On the renormalization of the Kardar-Parisi-Zhang equation

M. Lässig, Nucl. Phys. B448, 559, (1995)

The Kardar-Parisi-Zhang (KPZ) equation of nonlinear stochastic growth in d dimensions is studied using the mapping on to a system of directed polymers in a quenched random medium. The polymer problem is renormalized exactly in a minimally subtracted perturbation expansion about d=2. For the KPZ roughening transition in dimensions d>2, this renormalization group yields the dynamic exponent z⋆=2 and the roughness exponent χ⋆=0, which are exact to all orders in ε≡(2−d)/2. The expansion becomes singular in d=4, which is hence identiﬁed with the upper critical dimension of the KPZ equation. The implications of this perturbation theory for the strong-coupling phase are discussed. In particular, it is shown that the correlation functions and the coupling constant deﬁned in minimal subtraction develop an essential singularity at the strong-coupling ﬁxed point.

Selected applications to biology and bioinformatics

Significance analysis and statistical mechanics: an application to clustering

M. Luksza, M. Lässig, and J. Berg, Phys. Rev. Lett. 105, 220601 (4 pages), (2010)

This paper addresses the statistical significance of structures in random data: Given a set of vectors and a measure of mutual similarity, how likely does a subset of these vectors form a cluster with enhanced similarity among its elements? The computation of this cluster p-value for randomly distributed vectors is mapped onto a well-defined problem of statistical mechanics. We solve this problem analytically, establishing a connection between the physics of quenched disorder and multiple testing statistics in clustering and related problems. In an application to gene expression data, we find a remarkable link between the statistical significance of a cluster and the functional relationships between its genes.

Fitness flux and ubiquity of adaptive evolution

V. Mustonen and M. Lässig, Proc. Natl. Acad. Sci. 107, 4248-53, (2010)

Natural selection favors fitter variants in a population, but actual evolutionary processes may decrease fitness by mutations and genetic drift. How is the stochastic evolution of molecular biological systems shaped by natural selection? Here, we derive a theorem on the fitness flux in a population, defined as the selective effect of its genotype frequency changes. The fitness-flux theorem generalizes Fisher's fundamental theorem of natural selection to evolutionary processes including mutations, genetic drift, and time-dependent selection. It shows that a generic state of populations is adaptive evolution: there is a positive fitness flux resulting from a surplus of beneficial over deleterious changes. In particular, stationary nonequilibrium evolution processes are predicted to be adaptive. Under specific nonstationary conditions, notably during a decrease in population size, the average fitness flux can become negative. We show that these predictions are in accordance with experiments in bacteria and bacteriophages and with genomic data in Drosophila. Our analysis establishes fitness flux as a universal measure of adaptation in molecular evolution.

Molecular evolution under fitness fluctuations

V. Mustonen and M. Lässig, Phys Rev Lett. 100, 108101, (2008)

Molecular evolution is a stochastic process governed by fitness, mutations, and reproductive fluctuations in a population. Here, we study evolution where fitness itself is stochastic, with random switches in the direction of selection at individual genomic loci. As the correlation time of these fluctuations becomes larger than the diffusion time of mutations within the population, fitness changes from an annealed to a quenched random variable. We show that the rate of evolution has its maximum in the crossover regime, where both time scales are comparable. Adaptive evolution emerges in the quenched fitness regime (evidence for such fitness fluctuations has recently been found in genomic data). The joint statistical theory of reproductive and fitness fluctuations establishes a conceptual connection between evolutionary genetics and statistical physics of disordered system

Similarity detection and localization

T. Hwa and M. Lässig, Phys. Rev. Lett. 76, 2591, (1996)

The detection of similarities between long DNA and protein sequences is studied using concepts of statistical physics. It is shown that mutual similarities can be detected by sequence alignment methods only if their amount exceeds a threshold value. The onset of detection is a critical phase transition viewed as a localization-delocalization transition. The fidelity of the alignment is the order parameter of that transition; it leads to criteria to select optimal alignment parameters.

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