Thermodynamic implications of the gravitationally induced particle creation scenario

A rigorous thermodynamic analysis has been done as regards the apparent horizon of a spatially flat Friedmann–Lemaitre–Robertson–Walker universe for the gravitationally induced particle creation scenario with constant specific entropy and an arbitrary particle creation rate \(\Gamma \). Assuming a perfect fluid equation of state \(p=(\gamma -1)\rho \) with \(\frac{2}{3} \le \gamma \le 2\), the first law, the generalized second law (GSL), and thermodynamic equilibrium have been studied, and an expression for the total entropy (i.e., horizon entropy plus fluid entropy) has been obtained which does not contain \(\Gamma \) explicitly. Moreover, a lower bound for the fluid temperature \(T_f\) has also been found which is given by \(T_f \ge 8\left( \frac{\frac{3\gamma }{2}-1}{\frac{2}{\gamma }-1}\right) H^2\). It has been shown that the GSL is satisfied for \(\frac{\Gamma }{3H} \le 1\). Further, when \(\Gamma \) is constant, thermodynamic equilibrium is always possible for \(\frac{1}{2}<\frac{\Gamma }{3H} < 1\), while for \(\frac{\Gamma }{3H} \le \text {min}\left\{ \frac{1}{2},\frac{2\gamma -2}{3\gamma -2} \right\} \) and \(\frac{\Gamma }{3H} \ge 1\), equilibrium can never be attained. Thermodynamic arguments also lead us to believe that during the radiation phase, \(\Gamma \le H\). When \(\Gamma \) is not a constant, thermodynamic equilibrium holds if \(\ddot{H} \ge \frac{27}{4}\gamma ^2 H^3 \left( 1-\frac{\Gamma }{3H}\right) ^2\), however, such a condition is by no means necessary for the attainment of equilibrium.     

Thermodynamics phase transition and Hawking radiation of the Schwarzschild black hole with quintessence-like matter and a deficit solid angle

In this paper, we investigate the thermodynamics and Hawking radiation of Schwarzschild black hole with quintessence-like matter and deficit solid angle. From the metric of the black hole, we derive the expressions of temperature and specific heat using the laws of black hole thermodynamics. Using the null geodesics method and Parikh–Wilczeck tunneling method, we derive the expressions of Boltzmann factor and the change of Bekenstein–Hawking entropy for the black hole. The behaviors of the temperature, specific heat, Boltzmann factor and the change of Bekenstein entropy versus the deficit solid angle (\(\epsilon ^{2}\)) and the density of static spherically symmetric quintessence-like matter (\(\rho _{0}\)) were explicitly plotted. The results show that, when the deficit solid angle (\(\epsilon ^{2}\)) and the density of static spherically symmetric quintessence-like matter at \(r=1\) (\(\rho _{0}\)) vanish \((\rho _{0}=\epsilon =0)\), these four thermodynamics quantities are reduced to those obtained for the simple case of Schwarzschild black hole. For low entropies, the presence of quintessence-like matter induces a first order phase transition of the black hole and for the higher values of the entropies, we observe the second order phase transition. When increasing \(\rho _{0}\), the transition points are shifted to lower entropies. The same thing is observed when increasing \(\epsilon ^{2}\). In the absence of quintessence-like matter (\(\rho _{0}=0\)), these transition phenomena disappear. Moreover the rate of radiation decreases when increasing \(\rho _{0}\) or \((\epsilon ^2)\).     

Translation Invariant Extensions of Finite Volume Measures

We investigate the following questions: Given a measure \(\mu _\Lambda \) on configurations on a subset \(\Lambda \) of a lattice \(\mathbb {L}\), where a configuration is an element of \(\Omega ^\Lambda \) for some fixed set \(\Omega \), does there exist a measure \(\mu \) on configurations on all of \(\mathbb {L}\), invariant under some specified symmetry group of \(\mathbb {L}\), such that \(\mu _\Lambda \) is its marginal on configurations on \(\Lambda \)? When the answer is yes, what are the properties, e.g., the entropies, of such measures? Our primary focus is the case in which \(\mathbb {L}=\mathbb {Z}^d\) and the symmetries are the translations. For the case in which \(\Lambda \) is an interval in \(\mathbb {Z}\) we give a simple necessary and sufficient condition, local translation invariance (LTI), for extendibility. For LTI measures we construct extensions having maximal entropy, which we show are Gibbs measures; this construction extends to the case in which \(\mathbb {L}\) is the Bethe lattice. On \(\mathbb {Z}\) we also consider extensions supported on periodic configurations, which are analyzed using de Bruijn graphs and which include the extensions with minimal entropy. When \(\Lambda \subset \mathbb {Z}\) is not an interval, or when \(\Lambda \subset \mathbb {Z}^d\) with \(d>1\), the LTI condition is necessary but not sufficient for extendibility. For \(\mathbb {Z}^d\) with \(d>1\), extendibility is in some sense undecidable.     

A Theory of Hydrodynamic Turbulence Based on Non-equilibrium Statistical Mechanics

In earlier papers, we have studied the turbulent flow exponents \(\zeta _p\), where \(\langle |\Delta \mathbf{v}|^p\rangle \sim \ell ^{\zeta _p}\) and \(\Delta \mathbf{v}\) is the contribution to the fluid velocity at small scale \(\ell \). Using ideas of non-equilibrium statistical mechanics we have found

$$\begin{aligned} \zeta _p={p\over 3}-{1\over \ln \kappa }\ln \Gamma \left( {p\over 3}+1\right) \end{aligned}$$

where \(1/\ln \kappa \) is experimentally \(\approx \,0.32\,\pm \,0.01\). The purpose of the present note is to propose a somewhat more physical derivation of the formula for \(\zeta _p\). We also present an estimate \(\approx \,100\) for the Reynolds number at the onset of turbulence.

     

Thermodynamic variables of first-order entropy corrected Lovelock-AdS black holes: $$P{-}V$$ criticality analysis

We investigate the effect of thermal fluctuations on the thermodynamics of a Lovelock-AdS black hole. Taking the first order logarithmic correction term in entropy we analyze the thermodynamic potentials like Helmholtz free energy, enthalpy and Gibbs free energy. We find that all the thermodynamic potentials are decreasing functions of correction coefficient \(\alpha \). We also examined this correction coefficient must be positive by analysing \(P{-}V\) diagram. Further we study the \(P{-}V\) criticality and stability and find that presence of logarithmic correction in it is necessary to have critical points and stable phases. When \(P{-}V\) criticality appears, we calculate the critical volume \(V_c\), critical pressure \(P_c\) and critical temperature \(T_c\) using different equations and show that there is no critical point for this black hole without thermal fluctuations. We also study the geometrothermodynamics of this kind of black holes. The Ricci scalar of the Ruppeiner metric is graphically analysed.     

Effects of thermal fluctuations on non-minimal regular magnetic black hole

We analyze the effects of thermal fluctuations on a regular black hole (RBH) of the non-minimal Einstein–Yang–Mill theory with gauge field of magnetic Wu–Yang type and a cosmological constant. We consider the logarithmic corrected entropy in order to analyze the thermal fluctuations corresponding to non-minimal RBH thermodynamics. In this scenario, we develop various important thermodynamical quantities, such as entropy, pressure, specific heats, Gibb’s free energy and Helmholtz free energy. We investigate the first law of thermodynamics in the presence of logarithmic corrected entropy and non-minimal RBH. We also discuss the stability of this RBH using various frameworks such as the \(\gamma \) factor (the ratio of heat capacities), phase transition, grand canonical ensemble and canonical ensemble. It is observed that the non-minimal RBH becomes globally and locally more stable if we increase the value of the cosmological constant.     

Chern–Simons,Wess–Zumino and other cocycles from Kashiwara–Vergne and associators

Descent equations play an important role in the theory of characteristic classes and find applications in theoretical physics, e.g., in the Chern–Simons field theory and in the theory of anomalies. The second Chern class (the first Pontrjagin class) is defined as \(p= \langle F, F\rangle \) where F is the curvature 2-form and \(\langle \cdot , \cdot \rangle \) is an invariant scalar product on the corresponding Lie algebra \(\mathfrak g\). The descent for p gives rise to an element \(\omega =\omega _3+\omega _2+\omega _1+\omega _0\) of mixed degree. The 3-form part \(\omega _3\) is the Chern–Simons form. The 2-form part \(\omega _2\) is known as the Wess–Zumino action in physics. The 1-form component \(\omega _1\) is related to the canonical central extension of the loop group LG. In this paper, we give a new interpretation of the low degree components \(\omega _1\) and \(\omega _0\). Our main tool is the universal differential calculus on free Lie algebras due to Kontsevich. We establish a correspondence between solutions of the first Kashiwara–Vergne equation in Lie theory and universal solutions of the descent equation for the second Chern class p. In more detail, we define a 1-cocycle C which maps automorphisms of the free Lie algebra to one forms. A solution of the Kashiwara–Vergne equation F is mapped to \(\omega _1=C(F)\). Furthermore, the component \(\omega _0\) is related to the associator \(\Phi \) corresponding to F. It is surprising that while F and \(\Phi \) satisfy the highly nonlinear twist and pentagon equations, the elements \(\omega _1\) and \(\omega _0\) solve the linear descent equation.     

Summability of Connected Correlation Functions of Coupled Lattice Fields

We consider two nonindependent random fields \(\psi \) and \(\phi \) defined on a countable set Z. For instance, \(Z=\mathbb {Z}^d\) or \(Z=\mathbb {Z}^d\times I\), where I denotes a finite set of possible “internal degrees of freedom” such as spin. We prove that, if the cumulants of \(\psi \) and \(\phi \) enjoy a certain decay property, then all joint cumulants between \(\psi \) and \(\phi \) are \(\ell _2\)-summable in the precise sense described in the text. The decay assumption for the cumulants of \(\psi \) and \(\phi \) is a restricted \( \ell _1\) summability condition called \(\ell _1\)-clustering property. One immediate application of the results is given by a stochastic process \(\psi _t(x)\) whose state is \(\ell _1\)-clustering at any time t: then the above estimates can be applied with \(\psi =\psi _t\) and \(\phi =\psi _0\) and we obtain uniform in t estimates for the summability of time-correlations of the field. The above clustering assumption is obviously satisfied by any \(\ell _1\)-clustering stationary state of the process, and our original motivation for the control of the summability of time-correlations comes from a quest for a rigorous control of the Green–Kubo correlation function in such a system. A key role in the proof is played by the properties of non-Gaussian Wick polynomials and their connection to cumulants     

Finite-Size Effects in Non-neutral Two-Dimensional Coulomb Fluids

Thermodynamic potential of a neutral two-dimensional (2D) Coulomb fluid, confined to a large domain with a smooth boundary, exhibits at any (inverse) temperature \(\beta \) a logarithmic finite-size correction term whose universal prefactor depends only on the Euler number of the domain and the conformal anomaly number \(c=-1\). A minimal free boson conformal field theory, which is equivalent to the 2D symmetric two-component plasma of elementary \(\pm e\) charges at coupling constant \(\Gamma =\beta e^2\), was studied in the past. It was shown that creating a non-neutrality by spreading out a charge Qe at infinity modifies the anomaly number to \(c(Q,\Gamma ) = - 1 + 3\Gamma Q^2\). Here, we study the effect of non-neutrality on the finite-size expansion of the free energy for another Coulomb fluid, namely the 2D one-component plasma (jellium) composed of identical pointlike e-charges in a homogeneous background surface charge density. For the disk geometry of the confining domain we find that the non-neutrality induces the same change of the anomaly number in the finite-size expansion. We derive this result first at the free-fermion coupling \(\Gamma \equiv \beta e^2=2\) and then, by using a mapping of the 2D one-component plasma onto an anticommuting field theory formulated on a chain, for an arbitrary even coupling constant.     

Warm modified Chaplygin gas shaft inflation

In this paper, we examine the possible realization of a new inflation family called “shaft inflation” by assuming the modified Chaplygin gas model and a tachyon scalar field. We also consider the special form of the dissipative coefficient \(\Gamma ={a_0}\frac{T^{3}}{\phi ^{2 }}\) and calculate the various inflationary parameters in the scenario of strong and weak dissipative regimes. In order to examine the behavior of inflationary parameters, the \(n_s \)–\( \phi ,\, n_s \)–r, and \(n_s \)–\( \alpha _s\) planes (where \(n_s,\, \alpha _s,\, r\), and \(\phi \) represent the spectral index, its running, tensor-to-scalar ratio, and scalar field, respectively) are being developed, which lead to the constraints \(r< 0.11\), \(n_s=0.96 \pm 0.025\), and \(\alpha _s =-0.019 \pm 0.025\). It is quite interesting that these results of the inflationary parameters are compatible with BICEP2, WMAP \((7+9)\) and recent Planck data.     

Skeletal modifications of $$\upbeta $$-carboline alkaloids and their antiviral activity profile

To study the effect of the variation of fused ring size and substitution on the antiviral activity of \(\upbeta \)-carboline alkaloids, four types of structurally novel \(\upbeta \)-carboline alkaloids analogues, with indole-fused six- to nine-membered-rings motifs, were designed, synthesized, and evaluated for the inhibition of tobacco mosaic virus (TMV). Bioassay results indicated that most of these analogues had significant anti-TMV activity; especially I-14 (54 \(\pm \) 3 % at 500 \(\upmu \)g/mL in vitro; 51 \(\pm \) 2, 45 \(\pm \) 2, and 42 \(\pm \) 1 % at 500 \(\upmu \)g/mL in vivo), II-4 (53 \(\pm \) 1 % at 500 \(\upmu \)g/mL in vitro; 49 \(\pm \) 2, 57 \(\pm \) 2, and 48 \(\pm \) 1 % at 500 \(\upmu \)g/mL in vivo), and II-8 (48 \(\pm \) 1 % at 500 \(\upmu \)g/mL in vitro; 53 \(\pm \) 2 %, 56 \(\pm \) 2 %, and 46 \(\pm \) 1 % at 500 \(\upmu \)g/mL in vivo), which were more potent vs. TMV than was ribavirin (36 \(\pm \) 1 % at 500 \(\upmu \)g/mL in vitro; 37 \(\pm \) 2, 41 \(\pm \) 2, and 38 \(\pm \) 1 % at 500 \(\upmu \)g/mL in vivo). The size of the fused ring has important effects on anti-TMV potency, which may be ascribed to conformational differences. The X-ray structures of I-1, I-6, II-8, and III show differing conformational preferences. The most potent compounds can be used as leads for further optimization as antiphytoviral agents.     

Compound Poisson Law for Hitting Times to Periodic Orbits in Two-Dimensional Hyperbolic Systems

We show that a compound Poisson distribution holds for scaled exceedances of observables \(\phi \) uniquely maximized at a periodic point \(\zeta \) in a variety of two-dimensional hyperbolic dynamical systems with singularities \((M,T,\mu )\), including the billiard maps of Sinai dispersing billiards in both the finite and infinite horizon case. The observable we consider is of form \(\phi (z)=-\ln d(z,\zeta )\) where d is a metric defined in terms of the stable and unstable foliation. The compound Poisson process we obtain is a Pólya-Aeppli distibution of index \(\theta \). We calculate \(\theta \) in terms of the derivative of the map T. Furthermore if we define \(M_n=\max \{\phi ,\ldots ,\phi \circ T^n\}\) and \(u_n (\tau )\) by \(\lim _{n\rightarrow \infty } n\mu (\phi >u_n (\tau ) )=\tau \) the maximal process satisfies an extreme value law of form \(\mu (M_n \le u_n)=e^{-\theta \tau }\). These results generalize to a broader class of functions maximized at \(\zeta \), though the formulas regarding the parameters in the distribution need to be modified.     

The decays of $$\bar{B}^0$$, $$\bar{B}^0_s$$B¯s0 and $$B^-$$B- into $$\eta _c$$ηc plus a scalar or vector meson

We investigate the decays of \(\bar{B}^0_s\), \(\bar{B}^0\) and \(B^-\) into \(\eta _c\) plus a scalar or vector meson in a theoretical framework by taking into account the dominant process for the weak decay of \(\bar{B}\) meson into \(\eta _c\) and a \(q\bar{q}\) pair. After hadronization of this \(q\bar{q}\) component into pairs of pseudoscalar mesons we obtain certain weights for the pseudoscalar meson-pseudoscalar meson components. In addition, the \(\bar{B}^0\) and \(\bar{B}^0_s\) decays into \(\eta _c\) and \(\rho ^0\), \(K^*\) are evaluated and compared to the \(\eta _c\) and \(\phi \) production. The calculation is based on the postulation that the scalar mesons \(f_0(500)\), \(f_0(980)\) and \(a_0(980)\) are dynamically generated states from the pseudoscalar meson-pseudoscalar meson interactions in S-wave. Up to a global normalization factor, the \(\pi \pi \), \(K \bar{K}\) and \(\pi \eta \) invariant mass distributions for the decays of \(\bar{B}^0_s \rightarrow \eta _c \pi ^+ \pi ^-\), \(\bar{B}^0_s \rightarrow \eta _c K^+ K^-\), \(\bar{B}^0 \rightarrow \eta _c \pi ^+ \pi ^-\), \(\bar{B}^0 \rightarrow \eta _c K^+ K^-\), \(\bar{B}^0 \rightarrow \eta _c \pi ^0 \eta \), \(B^- \rightarrow \eta _c K^0 K^-\) and \(B^- \rightarrow \eta _c \pi ^- \eta \) are predicted. Comparison is made with the limited experimental information available and other theoretical calcualtions. Further comparison of these results with coming LHCb measurements will be very valuable to make progress in our understanding of the nature of the low lying scalar mesons, \(f_0(500), f_0(980)\) and \(a_0(980)\).     

Two-Higgs-doublet type-II and -III models and $$t\rightarrow c h$$ at the LHC

We study the constraints of the generic two-Higgs-doublet model (2HDM) type-III and the impacts of the new Yukawa couplings. For comparisons, we revisit the analysis in the 2HDM type-II. To understand the influence of all involving free parameters and to realize their correlations, we employ a \(\chi \)-square fitting approach by including theoretical and experimental constraints, such as the S, T, and U oblique parameters, the production of standard model Higgs and its decay to \(\gamma \gamma \), \(WW^*/ZZ^*\), \(\tau ^+\tau ^-\), etc. The errors of the analysis are taken at 68, 95.5, and \(99.7~\%\) confidence levels. Due to the new Yukawa couplings being associated with \(\cos (\beta -\alpha )\) and \(\sin (\beta -\alpha )\), we find that the allowed regions for \(\sin \alpha \) and \(\tan \beta \) in the type-III model can be broader when the dictated parameter \(\chi _F\) is positive; however, for negative \(\chi _F\), the limits are stricter than those in the type-II model. By using the constrained parameters, we find that the deviation from the SM in \(h\rightarrow Z\gamma \) can be of \(\mathcal{O}(10~\%)\). Additionally, we also study the top-quark flavor-changing processes induced at the tree level in the type-III model and find that when all current experimental data are considered, we get \(Br(t\rightarrow c(h, H) )< 10^{-3}\) for \(m_h=125.36\) and \(m_H=150\) GeV, and \(Br(t\rightarrow cA)\) slightly exceeds \(10^{-3}\) for \(m_A =130\) GeV.     

Equilibration in the Kac Model Using the GTW Metric $$d_2$$

We use the Fourier based Gabetta–Toscani–Wennberg metric \(d_2\) to study the rate of convergence to equilibrium for the Kac model in 1 dimension. We take the initial velocity distribution of the particles to be a Borel probability measure \(\mu \) on \(\mathbb {R}^n\) that is symmetric in all its variables, has mean \(\vec {0}\) and finite second moment. Let \(\mu _t(dv)\) denote the Kac-evolved distribution at time t, and let \(R_\mu \) be the angular average of \(\mu \). We give an upper bound to \(d_2(\mu _t, R_\mu )\) of the form \(\min \left\{ B e^{-\frac{4 \lambda _1}{n+3}t}, d_2(\mu ,R_\mu )\right\} ,\) where \(\lambda _1 = \frac{n+2}{2(n-1)}\) is the gap of the Kac model in \(L^2\) and B depends only on the second moment of \(\mu \). We also construct a family of Schwartz probability densities \(\{f_0^{(n)}: \mathbb {R}^n\rightarrow \mathbb {R}\}\) with finite second moments that shows practically no decrease in \(d_2(f_0(t), R_{f_0})\) for time at least \(\frac{1}{2\lambda }\) with \(\lambda \) the rate of the Kac operator. We also present a propagation of chaos result for the partially thermostated Kac model in Tossounian and Vaidyanathan (J Math Phys 56(8):083301, 2015).     

Optimisation of a hybrid photoneutron source in a linear accelerator using GEANT4 and MCNPX Monte Carlo codes

The \(\alpha \) decay half-lives of hyper and normal isotopes of Po nuclei are studied in the present work. The inclusion of \(\Lambda \)–N interaction changes the half-life for \(\alpha \) decay. The theoretical predictions on the \(\alpha \) decay half-lives of normal Po isotopes are compared with experimental results and are seen to be matching well with each other. The neutron shell closure at \(N = 126\) is found to be the same for both normal and hypernuclei. The Geiger–Nuttal (G–N) law for \(\alpha \) decay is unaltered in the case of hypernuclei. The hypernuclei will decay into normal nuclei by mesonic or non-mesonic decay modes. Since the half-lives of normal Po nuclei are well within the experimental limits, our theoretical results suggest experimental verification of the \(\alpha \) emission from hyper Po nuclei in a cascade process.     

Regularity of the Speed of Biased Random Walk in a One-Dimensional Percolation Model

We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelson-Fisk and Häggström established for this model a phase transition for the asymptotic linear speed \(\overline{\hbox {v}}\) of the walk. Namely, there exists some critical value \(\lambda _{\hbox {c}}>0\) such that \(\overline{\hbox {v}}>0\) if \(\lambda \in (0,\lambda _{\hbox {c}})\) and \(\overline{\hbox {v}}=0\) if \(\lambda \ge \lambda _{\hbox {c}}\). We show that the speed \(\overline{\hbox {v}}\) is continuous in \(\lambda \) on \((0,\infty )\) and differentiable on \((0,\lambda _{\hbox {c}}/2)\). Moreover, we characterize the derivative as a covariance. For the proof of the differentiability of \(\overline{\hbox {v}}\) on \((0,\lambda _{\hbox {c}}/2)\), we require and prove a central limit theorem for the biased random walk. Additionally, we prove that the central limit theorem fails to hold for \(\lambda \ge \lambda _{\hbox {c}}/2\).     

Absorption of scalars by extremal black holes in string theory

We show that the low frequency absorption cross section of minimally coupled test massless scalar fields by extremal spherically symmetric black holes in d dimensions is equal to the horizon area, even in the presence of string-theoretical \(\alpha '\) corrections. Classically one has the relation \(\sigma = 4 GS\) between that absorption cross section and the black hole entropy. By comparing in each case the values of the horizon area and Wald’s entropy, we discuss the validity of such relation in the presence of higher derivative corrections for extremal black holes in many different contexts: in the presence of electric and magnetic charges; for nonsupersymmetric and supersymmetric black holes; in \(d=4\) and \(d=5\) dimensions. The examples we consider seem to indicate that this relation is not verified in the presence of \(\alpha '\) corrections in general, although being valid in some specific cases (electrically charged maximally supersymmetric black holes in \(d=5\)). We argue that the relation \(\sigma = 4 GS\) should in general be valid for the absorption cross section of scalar fields which, although being independent from the black hole solution, have their origin from string theory, and therefore are not minimally coupled.     

Logarithmic Finite-Size Correction in Non-neutral Two-Component Plasma on Sphere

We consider a general two-component plasma of classical pointlike charges \(+e\) (e is say the elementary charge) and \(-Z e\) (valency \(Z=1,2,\ldots \)), living on the surface of a sphere of radius R. The system is in thermal equilibrium at the inverse temperature \(\beta \), in the stability region against collapse of oppositely charged particle pairs \(\beta e^2 < 2/Z\). We study the effect of the system excess charge Qe on the finite-size expansion of the (dimensionless) grand potential \(\beta \varOmega \). By combining the stereographic projection of the sphere onto an infinite plane, the linear response theory and the planar results for the second moments of the species density correlation functions we show that for any \(\beta e^2 < 2/Z\) the large-R expansion of the grand potential is of the form \(\beta \varOmega \sim A_V R^2 + \left[ \chi /6 - \beta (Qe)^2/2\right] \ln R\), where \(A_V\) is the non-universal coefficient of the volume (bulk) part and the Euler number of the sphere \(\chi =2\). The same formula, containing also a non-universal surface term proportional to R, was obtained previously for the disc domain (\(\chi =1\)), in the case of the symmetric \((Z=1)\) two-component plasma at the collapse point \(\beta e^2=2\) and the jellium model \((Z\rightarrow 0)\) of identical e-charges in a fixed neutralizing background charge density at any coupling \(\beta e^2\) being an even integer. Our result thus indicates that the prefactor to the logarithmic finite-size expansion does not depend on the composition of the Coulomb fluid and its non-universal part \(-\beta (Qe)^2/2\) is independent of the geometry of the confining domain.     

Gibbs Measures Over Locally Tree-Like Graphs and Percolative Entropy Over Infinite Regular Trees

Consider a statistical physical model on the d-regular infinite tree \(T_{d}\) described by a set of interactions \(\Phi \). Let \(\{G_{n}\}\) be a sequence of finite graphs with vertex sets \(V_n\) that locally converge to \(T_{d}\). From \(\Phi \) one can construct a sequence of corresponding models on the graphs \(G_n\). Let \(\{\mu _n\}\) be the resulting Gibbs measures. Here we assume that \(\{\mu _{n}\}\) converges to some limiting Gibbs measure \(\mu \) on \(T_{d}\) in the local weak\(^*\) sense, and study the consequences of this convergence for the specific entropies \(|V_n|^{-1}H(\mu _n)\). We show that the limit supremum of \(|V_n|^{-1}H(\mu _n)\) is bounded above by the percolative entropy \(H_{\textit{perc}}(\mu )\), a function of \(\mu \) itself, and that \(|V_n|^{-1}H(\mu _n)\) actually converges to \(H_{\textit{perc}}(\mu )\) in case \(\Phi \) exhibits strong spatial mixing on \(T_d\). When it is known to exist, the limit of \(|V_n|^{-1}H(\mu _n)\) is most commonly shown to be given by the Bethe ansatz. Percolative entropy gives a different formula, and we do not know how to connect it to the Bethe ansatz directly. We discuss a few examples of well-known models for which the latter result holds in the high temperature regime.