Concepts in Thermal Physics

Blundell, Stephen J.; Blundell, Katherine M.

doi:10.1093/acprof:oso/9780199562091.001.0001

Cited by 233 publications

(311 citation statements)

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“…If one identifies the Fiedler eigenvalue µ N −1 with the internal energy of a thermodynamical system, then its first derivative exhibits a jump from zero to a finite value. Nevertheless, this derivative does not diverge as expected for a second order 1 transition [33]. On the other hand, if one employs the Fiedler eigenvalue as a metric for the synchronizability and regards it as a thermodynamical potential such as the free enthalpy, its Legendre transform corresponds to the internal energy and exhibits a discontinuity at α = α I .…”

Section: Physical Interpretationmentioning

confidence: 76%

Algebraic connectivity of interdependent networks

Martı́n-Hernández

Wang

Mieghem

et al. 2014

Physica A: Statistical Mechanics and its Applications

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The algebraic connectivity µ N −1 , i.e. the second smallest eigenvalue of the Laplacian matrix, plays a crucial role in dynamic phenomena such as diffusion processes, synchronization stability, and network robustness. In this work we study the algebraic connectivity in the general context of interdependent networks, or network-of-networks (NoN). The present work shows, both analytically and numerically, how the algebraic connectivity of NoNs experiences a transition. The transition is characterized by a saturation of the algebraic connectivity upon the addition of sufficient coupling links (between the two individual networks of a NoN). In practical terms, this shows that NoN topologies require only a fraction of coupling links in order to achieve optimal diffusivity. Furthermore, we observe a footprint of the transition on the properties of Fiedler's spectral bisection.

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Section: Physical Interpretationmentioning

confidence: 76%

Algebraic connectivity of interdependent networks

Martı́n-Hernández

Wang

Mieghem

et al. 2014

Physica A: Statistical Mechanics and its Applications

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“…For gases, the transport coefficients for the transport of heat and momentum are nearly equal and the Prandtl number is of order unity, with Pr = 2 3 when γ = 5/3 (Blundell & Blundell 2006). This suggests that a constraint on the particle AC parameter α C i can be obtained by setting…”

Section: The Blob Testmentioning

confidence: 99%

Hydrodynamic capabilities of an SPH code incorporating an artificial conductivity term with a gravity-based signal velocity

Valdarnini

2012

A&A

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This paper investigates the hydrodynamic performances of an smoothed particle hydrodynamics (SPH) code incorporating an artificial heat conductivity term in which the adopted signal velocity is applicable when gravity is present. To this end, we analyze results from simulations produced using a suite of standard hydrodynamical test problems. In accordance with previous findings, we show that the performances of SPH in describing the development of Kelvin-Helmholtz instabilities depend strongly on both the consistency of the initial condition set-up and the leading error in the momentum equation due to incomplete kernel sampling. In contrast, the presence of artificial conductivity does not significantly affect the results. An error and stability analysis shows that the quartic B-spline kernel (M 5 ) possesses very good stability properties and so we propose its use with a large neighbor number, between ∼50 (2D) to ∼100 (3D), to improve convergence in simulation results without being affected by the so-called clumping instability. Moreover, the results of the Sod shock tube demonstrate that to obtain simulation profiles in accord with the analytic solution, for simulations employing kernels with a non-zero first derivative at the origin, it is necessary to use a much larger number of neighbors than in the case of the M 5 runs. Our SPH simulations of the blob test show that in order to achieve blob disruption it is necessary to include an artificial conductivity term. However, we find that in the regime of strong supersonic flows an appropriate limiting condition, which depends on the Prandtl number, must be imposed on the artificial conductivity SPH coefficients in order to avoid an unphysical amount of heat diffusion. Our results from hydrodynamic simulations that include self-gravity show profiles of hydrodynamic variables that are in much better agreement with those produced using mesh-based codes. In particular, the final levels of core entropies in cosmological simulations of galaxy clusters are consistent with those found using AMR codes. This demonstrates that the proposed diffusion scheme is capable of mimicking the process of entropy mixing that is produced during structure formation because of the diffusion caused by turbulence. Finally, the results of our Rayleigh-Taylor instability test demonstrate that in the regime of very subsonic flows the code still has several difficulties in the treatment of hydrodynamic instabilities. These problems are intrinsic to the way in which standard SPH gradients are calculated and not to the implementation of the artificial conductivity term. To overcome these difficulties, several numerical schemes have been proposed that, if coupled with the SPH implementation presented in this paper, could solve the issues that have recently been addressed in investigating SPH performances to model subsonic turbulence.

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“…These include, among others, the energy associated with the Ising spin models 26 and cost functions of combinatorial optimization problems, such as the travelling salesman problem 27 . In summary, the structural transition of interdependent networks involves a discontinuity in the first derivative of an energy-like function, and thus, according to the Ehrenfest classification of phase transitions, it is a discontinuous transition 28 .…”

mentioning

confidence: 99%

Abrupt transition in the structural formation of interconnected networks

2013

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Our world is linked by a complex mesh of networks through which information, people and goods flow. These networks are interdependent on each other, and present structural and dynamical features 1-6 different from those observed in isolated networks 7-9 . Although examples of such dissimilar properties are becoming more abundant-such as in diffusion, robustness and competition-it is not yet clear where these differences are rooted. Here we show that the process of building independent networks into an interconnected network of networks undergoes a structurally sharp transition as the interconnections are formed. Depending on the relative importance of inter-and intra-layer connections, we find that the entire interdependent system can be tuned between two regimes: in one regime, the various layers are structurally decoupled and they act as independent entities; in the other regime, network layers are indistinguishable and the whole system behaves as a singlelevel network. We analytically show that the transition between the two regimes is discontinuous even for finite-size networks. Thus, any real-world interconnected system is potentially at risk of abrupt changes in its structure, which may manifest new dynamical properties.Interacting, interdependent or multiplex networks are different ways of naming the same class of complex systems where networks are not considered as isolated entities but interacting with each other. In multiplex, the nodes at each network are instances of the same entity; thus, the networks are representing simply different categorical relationships between entities, and usually categories are represented by layers. Interdependent networks is a more general framework where nodes can be different at each network.Many, if not all, real networks are coupled with other real networks. Examples can be found in several domains: social networks (for example, Facebook, Twitter and so on) are coupled because they share the same actors 10 ; multimodal transportation networks are composed of different layers (for example, bus, subway and so on) that share the same locations 11 ; the functioning of communication and power grid systems depends one on the other 1 . So far, all phenomena that have been studied on interdependent networks, including percolation 1,3 , epidemics 4 and linear dynamical systems 5 , have provided results that differ much from those valid in the case of isolated complex networks. Sometimes the difference is radical: for example, whereas isolated scale-free networks are robust against failures of their nodes or edges 12 , scale-free interdependent networks are instead very fragile 1,3 .Given such observations, two fundamentally important theoretical questions are in order: Why do dynamical and critical phenomena running on interdependent network models differ

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Concepts in Thermal Physics

Cited by 233 publications

References 0 publications

Algebraic connectivity of interdependent networks

Algebraic connectivity of interdependent networks

Hydrodynamic capabilities of an SPH code incorporating an artificial conductivity term with a gravity-based signal velocity

Abrupt transition in the structural formation of interconnected networks

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