In statistical physics any given system can be either at an equilibrium or away from it. Networks are not an exception. Most network models can be classified as either equilibrium or growing. Here we show that under certain conditions there exists an equilibrium formulation for any growing network model, and vice versa. The equivalence between the equilibrium and nonequilibrium formulations is exact not only asymptotically, but even for any finite system size. The required conditions are satisfied in random geometric graphs in general and causal sets in particular, and to a large extent in some real networks.
A large family of diffusive models of transport that has been considered in the past years admits a transformation into the same model in contact with an equilibrium bath. This mapping holds at the full dynamical level, and is independent of dimension or topology. It provides a good opportunity to discuss questions of time reversal in out of equilibrium contexts. In particular, thanks to the mapping one may define the free-energy in the non-equilibrium states very naturally as the (usual) free energy of the mapped system.
Velocity correlations in a 2D granular fluid are studied in experiments and numerical simulations. The transverse component of the velocity structure factor reveals two well defined energy scales, associated with the external "bath temperature" $T_b$ and with the internal granular one, $T_g<T_b$, relevant at large and small wavelengths respectively. Experimental and numerical data are discussed within a fluctuating hydrodynamics model, which allows one to define and measure a non-equilibrium coherence length $ξ$, growing with density, that characterizes order in the velocity field.
Intertwining Markov Processes via Matrix Product Operators
University of Modena and Reggio Emilia（摩德纳和雷焦艾米利亚大学）；INFN, Sezione di Bologna（意大利国家核物理研究所博洛尼亚分部）；The University of Melbourne（墨尔本大学）；University of Cambridge（剑桥大学）；Ghent University（根特大学）
Understanding the role of non-equilibrium driving in self-organization is crucial for developing a predictive description of biological systems, yet it is impeded by their complexity. The actin cytoskeleton serves as a paradigm for how equilibrium and non-equilibrium forces combine to give rise to self-organization. Motivated by recent experiments that show that actin filament growth rates can tune the morphology of a growing actin bundle crosslinked by two competing types of actin binding proteins, we construct a minimal model for such a system and show that the dynamics are subject to a set …
New entropy measures have been recently introduced for the quantification of the complexity of networks. Most of these entropy measures apply to static networks or to dynamical processes defined on static complex networks. In this paper we define the entropy rate of growing network models. This entropy rate quantifies how many labeled networks are typically generated by the growing network models. We analytically evaluate the difference between the entropy rate of growing tree network models and the entropy of tree networks that have the same asymptotic degree distribution. We find that the gr…
We point out a remarkable analogy between the Jarzynski identity from non-equilibrium statistical physics and the AdS/CFT duality. We apply the logic that leads to the Jarzynski identity to renormalization group (RG) flows of quantum field theories and then argue for the natural connection with the AdS/CFT duality formula. This application can be in principle checked in Monte Carlo simulations of RG flows. Given the existing generalizations of the Jarzynski identity in non-equilibrium statistical physics, and the analogy between the Jarzynski identity and the AdS/CFT duality, we are led to sug…
We point out that the recent discussion of non-relativistic AdS/CFT correspondence has a direct application in non-equilibrium statistical physics, the fact which has not been emphasized in the recent literature on the subject. In particular, we propose a duality between aging in systems far from equilibrium characterized by the dynamical exponent $z=2$ and gravity in a specific background. The key ingredient in our proposal is the recent geometric realization of the Schrödinger group. We also discuss the relevance of the proposed correspondence for the more general aging phenomena in systems …
We consider some duality relations for models of non-interacting particles hopping on disordered one-dimensional chains. In particular, we discuss symmetries of bulk-driven barrier and trap models, and relations between boundary-driven and equilibrium models with related energy landscapes. We discuss the relationships between these duality relations and similar results for interacting many-body systems.
The irreversible growth of magnetic films is studied in three-dimensional confined geometries of size $L\times L\times M$, where $M\gg L$ is the growing direction. Competing surface magnetic fields, applied to opposite corners of the growing system, lead to the observation of a localization-delocalization (weakly rounded) transition of the interface between domains of up and down spins on the planes transverse to the growing direction. This effective transition is the precursor of a true far-from-equilibrium corner wetting transition that takes place in the thermodynamic limit. The phenomenon …
We examine the duality relating the equilibrium dynamics of the mean-field $p$-spin ferromagnets at finite size in the Guerra's interpolation scheme and the Burgers hierarchy. In particular, we prove that - for fixed $p$ - the expectation value of the order parameter on the first side w.r.t. the generalized partition function satisfies the $p-1$-th element in the aforementioned class of nonlinear equations. In the light of this duality, we interpret the phase transitions in the thermodynamic limit of the statistical mechanics model with the development of shock waves in the PDE side. We also o…
There is a complex relation between the mechanism of preferential attachment, scale-free degree distributions and hyperbolicity in complex networks. In fact, both preferential attachment and hidden hyperbolic spaces often generate scale-free networks. We show that there is actually a duality between a class of growing spatial networks based on preferential attachment on the sphere and a class of static random networks on the hyperbolic plane. Both classes of networks have the same scale-free degree distribution as the Barabasi-Albert model. As a limit of this correspondence, the Barabasi-Alber…
Let $X$ be a finite set and $Ω=\{1,...,d\}^{\mathbb{N}}$ be the Bernoulli space. Denote by $σ$ the shift map acting on $Ω$. For a fixed probability $μ$ on $X$ with supp($μ$)$=X$, define $Π(μ,σ)$ as the set of all Borel probabilities $π\in P(X\times Ω)$ such that the $x$-marginal of $π$ is $μ$ and the $y$-marginal of $π$ is $σ$-invariant. We consider a fixed Lipschitz cost function $c: X \times Ω\to \mathbb{R}$ and an associated Ruelle operator. We introduce the concept of Gibbs plan, which is a probability on $X \times Ω$. Moreover, we define entropy, pressure and equilibrium plans. The study …
By embedding inert tracer particles (TPs) in a growing multicellular spheroid the local stresses on the cancer cells (CCs) can be measured. In order for this technique to be effective the unknown effect of the dynamics of the TPs on the CCs has to be elucidated to ensure that the TPs do not greatly alter the local stresses on the CCs. We show, using theory and simulations, that the self-generated (active) forces arising from proliferation and apoptosis of the CCs drive the dynamics of the TPs far from equilibrium. On time scales less than the division times of the CCs, the TPs exhibit sub-diff…
1. Basic constructions. 2. Equilibrium and nonequilibrium networks. 3. Equilibrium uncorrelated networks. 4. Nonequilibrium nongrowing scale-free nets. 5. Types of correlations. 6. When pair correlations are important. 7. When loops are important. 8. Pair degree-degree correlations in growing networks. 9. How to construct an equilibrium net with given degree-degree correlations. 10. How to construct a growing scale-free net with a given clustering (towards a real-space renormalization group for networks).
The Barabási-Bianconi (BB) fitness model can be solved by a mapping between the original network growth model to an idealized bosonic gas. The well-known transition to Bose-Einstein condensation in the latter then corresponds to the emergence of "super-hubs" in the network model. Motivated by the preservation of the scale-free property, thermodynamic stability and self-duality, we generalize the original extensive mapping of the BB fitness model by using the nonextensive Kaniadakis k-distribution. Through numerical simulation and mean-field calculations we show that deviations from extensivity…
The non-equilibrium dynamics of three paradigmatic models for two-dimensional systems with quenched disorder is studied with a focus on the existence and analysis of a growing length scale during aging at low temperatures: 1) The random bond Ising ferromagnet, 2) the Edwards-Anderson model for a spin glas, 3) the solid-on-solid model on a disordered substrate (equivalent to the sine-Gordon model with random phase shifts). Interestingly, we find in all three models a length scale that grows algebraically with time (up to the system size in cases 1 and 3, up to the finite equilibrium length in c…
Simplicial complexes describe collaboration networks, protein interaction networks and brain networks and in general network structures in which the interactions can include more than two nodes. In real applications, often simplicial complexes are weighted. Here we propose a non-equilibrium model for weighted growing simplicial complexes. The proposed dynamics is able to generate weighted simplicial complexes with a rich interplay between weights and topology emerging not just at the level of nodes and links, but also at the level of faces of higher dimension.
We describe an ensemble of growing scale-free networks in an equilibrium framework, providing insight into why the exponent of empirical scale-free networks in nature is typically robust. In an analogy to thermostatistics, to describe the canonical and microcanonical ensembles, we introduce a functional, whose maximum corresponds to a scale-free configuration. We then identify the equivalents to energy, Zeroth-law, entropy and heat capacity for scale-free networks. Discussing the merging of scale-free networks, we also establish an exact relation to predict their final "equilibrium" degree exp…
The growth of confined magnetic films with ferromagnetic interactions between nearest-neighbor spins is studied in a stripped $(1+1)-$dimensional rectangular geometry. Magnetic films are grown irreversibly by adding spins at the boundaries of the growing interface. A competing situation with two opposite short range surface magnetic fields of the same magnitude is analyzed. Due to the antisymmetric condition considered, an interface between domains with spins having opposite orientations develops along the growing direction. Such interface undergoes a localization-delocalization transition tha…
Magnetic Eden clusters with ferromagnetic interaction between nearest-neighbor spins are grown in a confined 2d-geometry with short range magnetic fields acting on the surfaces. The change of the growing interface curvature driven by the field and the temperature is identified as a non-equilibrium wetting transition and the corresponding phase diagram is evaluated.
We study the geometric properties of a system initially in equilibrium at a critical point that is suddenly quenched to another critical point and subsequently evolves towards the new equilibrium state. We focus on the bidimensional Ising model and we use numerical methods to characterize the morphological and statistical properties of spin and Fortuin-Kasteleyn clusters during the critical evolution. The analysis of the dynamics of an out of equilibrium interface is also performed. We show that the small scale properties, smaller than the target critical growing length $ξ(t) \sim t^{1/z}$ wit…
Scale-free (SF) networks and small world networks have been found to occur in very diverse contexts. It is this striking universality which makes one look for widely applicable mechanisms which lead to the formation of such networks. In this letter we propose a new mechanism for the construction of SF networks: Evolving networks as interaction networks of systems which are distinguished by their stability if perturbed out of equilibrium. Stability is measured by the largest real part of any eigenvalue of a matrix associated with the graph. We extend the model to weighted directed networks and …
A stochastic approach to the filling dynamics of an open topology porous structure permeated with a perfectly wetting fluid is presented. From the discrete structure of the disordered voids network with only nearest neighbors links, we derive the "microscopic" (at the pores scale) dynamical equations governing the filling dynamics of the coupled pores and the fluid pressure dynamics. The model yields two fundamental consequences. The first consequence regards the emergence of Darcy's law and the dependence of the predicted permeability with the voids network topology. The second one is the pre…
The irreversible growth of a binary mixture under far-from-equilibrium conditions is studied in three-dimensional confined geometries of size $L_x \times L_y \times L_z$, where $L_z \gg L_x = L_y$ is the growing direction. A competing situation where two opposite surfaces prefer different species of the mixture is analyzed. Due to this antisymmetric condition an interface between the different species develops along the growing direction. Such interface undergoes a localization-delocalization transition that is the precursor of a wetting transition in the thermodynamic limit. Furthermore, the …
