Stretching Generic Pesin’s Entropy Formula

Journal of Statistical Physics - We prove that Pesin’s entropy formula holds generically within a broad subset of volume-preserving bi-Lipschitz homeomorphisms with respect to the...     

Properties of Segal’s Entropy for Quantum Systems

Properties of Segal’s entropy for semifinite and finite von Neumann algebras are investigated. In particular, its invariance with respect to a trace-preserving normal *-homomorphism is studied, as well as norm-continuity in the trace norm on the set of bounded in the operator norm density matrices.     

Rate of Convergence for Cardy’s Formula

We show that crossing probabilities in 2D critical site percolation on the triangular lattice in a piecewise analytic Jordan domain converge with power law rate in the mesh size to their limit given by the Cardy–Smirnov formula. We use this result to obtain new upper and lower bounds of ${e^{O(\sqrt{{\rm log}\,{\rm log} R})}\, R^{-1/3}}$ for the probability that the cluster at the origin in the half-plane has diameter R, improving the previously known estimate of R ^?1/3+o(1).     

Bell’s Correlations and Spin Systems

The structure of maximal violators of Bell’s inequalities for Jordan algebras is investigated. It is proved that the spin factor V ₂ is responsible for maximal values of Bell’s correlations in a faithful state. In this situation maximally correlated subsystems must overlap in a nonassociative subalgebra. For operator commuting subalgebras it is shown that maximal violators have the structure of the spin systems and that the global state (faithful on local subalgebras) acts as the trace on local subalgebras.     

Prospects for constrained supersymmetry at $$\sqrt{s}={33}\,\text {TeV} $$ and $$\sqrt{s}={100}\,\text {TeV} $$ proton–proton super-colliders

Successful models of pure gravity mediation (PGM) with radiative electroweak symmetry breaking can be expressed with as few as two free parameters, which can be taken as the gravitino mass and \(\tan \beta \) . These models easily support a 125–126 GeV Higgs mass at the expense of a scalar spectrum in the multi-TeV range and a much lighter wino as the lightest supersymmetric particle. In these models, it is also quite generic that the Higgs mixing mass parameter, \(\mu \) , which is determined by the minimization of the Higgs potential is also in the multi-TeV range. For \(\mu >0\) , the thermal relic density of winos is too small to account for the dark matter. The same is true for \(\mu <0\) unless the gravitino mass is of order 500 TeV. Here, we consider the origin of a multi-TeV \(\mu \) parameter arising from the breakdown of a Peccei–Quinn (PQ) symmetry. A coupling of the PQ-symmetry breaking field, \(P\) , to the MSSM Higgs doublets, naturally leads to a value of \(\mu \sim \langle P \rangle ^2 /M_P \sim {\mathcal O}(100)\) TeV and of the order that is required in PGM models. In this case, axions make up the dark matter or some fraction of the dark matter with the remainder made up from thermal or non-thermal winos. We also provide solutions to the problem of isocurvature fluctuations with axion dark matter in this context.     

Dirac multimode ket-bra operators’ \mathfrak{Q}-ordered and \mathfrak{P}-ordered integration theory and general squeezing operator

We develop quantum mechanical Dirac ket-bra operator’s integration theory in $\mathfrak{Q}$ -ordering or $\mathfrak{P}$ -ordering to multimode case, where $\mathfrak{Q}$ -ordering means all Qs are to the left of all Ps and $\mathfrak{P}$ -ordering means all Ps are to the left of all Qs. As their applications, we derive $\mathfrak{Q}$ -ordered and $\mathfrak{P}$ -ordered expansion formulas of multimode exponential operator $e^{ - iP_l \Lambda _{lk} Q_k } $ . Application of the new formula in finding new general squeezing operators is demonstrated. The general exponential operator for coordinate representation transformation $\left| {\left. {\left( {_{q_2 }^{q_1 } } \right)} \right\rangle \to } \right|\left. {\left( {_{CD}^{AB} } \right)\left( {_{q_2 }^{q_1 } } \right)} \right\rangle $ is also derived. In this way, much more correpondence relations between classical coordinate transformations and their quantum mechanical images can be revealed.     

Semi-Classical Twists for {\mathfrak{sl}}_{3} and {\mathfrak{sl}}_{4} Boundary r-Matrices of Cremmer–Gervais Type

We obtain explicit formulas for the semi-classical twists deforming the coalgebraic structure of $U({\mathfrak{sl}}_{3})$ and $U({\mathfrak{sl}}_{4})$ . In rank 2 and 3 the corresponding universal R-matrices quantize the boundary r-matrices of Cremmer–Gervais type defining Lie Frobenius structures on the maximal parabolic subalgebras in ${\mathfrak{sl}}_{n}$ .     

Entropy production and fluctuations in a Maxwell’s refrigerator with squeezing

Clarifying the impact of quantumness in the operation and properties of thermal machines represents a major challenge. Here we envisage a toy model acting either as an information-driven fridge or as heat-powered information eraser in which coherences can be naturally introduced in by means of squeezed thermal reservoirs. We study the validity of the transient entropy production fluctuation theorem in the model with and without squeezing as well as its decomposition into adiabatic and non-adiabatic contributions. Squeezing modifies fluctuations and introduces extra mechanisms of entropy exchange. This leads to enhancements in the cooling performance of the refrigerator, and challenging Landauer’s bound in the setup.     

Kolmogorov’s Theory of Turbulence and Inviscid Limit of the Navier-Stokes Equations in $${\mathbb {R}^3}$$

We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations in \mathbb R³{\mathbb {R}^3} . We first observe that a pathwise Kolmogorov hypothesis implies the uniform boundedness of the α ^th -order fractional derivatives of the velocity for some α > 0 in the space variables in L ², which is independent of the viscosity μ > 0. Then it is shown that this key observation yields the L ²-equicontinuity in the time variable and the uniform bound in L ^q , for some q > 2, of the velocity independent of μ > 0. These results lead to the strong convergence of solutions of the Navier-Stokes equations to a solution of the Euler equations in \mathbb R³{\mathbb {R}^3} . We also consider passive scalars coupled to the incompressible Navier-Stokes equations and, in this case, find the weak-star convergence for the passive scalars with a limit in the form of a Young measure (pdf depending on space and time). Not only do we offer a framework for mathematical existence theories, but also we offer a framework for the interpretation of numerical solutions through the identification of a function space in which convergence should take place, with the bounds that are independent of μ > 0, that is in the high Reynolds number limit.     

Search for chargino and neutralino production at \sqrt{s} = 181–184 GeV at LEP

A search for charginos and neutralinos, predicted by supersymmetric theories, has been performed using a data sample of 57 pb at centre-of-mass energies of 181–184 GeV taken with the OPAL detector at LEP. No evidence for chargino or neutralino production has been found. Upper limits on chargino and neutralino pair production (, ) cross-sections are obtained as a function of the chargino mass (), the lightest neutralino mass () and the second lightest neutralino mass (). For large chargino masses the limits have been improved with respect to the previous analyses at lower centre-of-mass energies. Exclusion regions at 95% confidence level (C.L.) of parameters of the Constrained Minimal Supersymmetric Standard Model are determined for the case of a large universal scalar mass, , implying heavy scalar fermions, and for the case of a small resulting in light scalar fermions and giving the worst-case limits. Within this framework and for GeV the 95% C.L. lower limits on for GeV are 90.0 and 90.2 GeV for and 35 respectively. These limits for all (the worst-case) are 69.1 and 65.2 GeV for and 35 respectively. Exclusion regions are also presented for neutralino masses, including an absolute lower limit at 95% C.L. for the mass of the lightest neutralino of 30.1 GeV for GeV (24.2 GeV for all ), with implications for experimental searches for the lightest neutralino as a dark matter candidate. Received: 19 August 1998 / Published online: 11 March 1999     

On Gammelgaard’s Formula for a Star Product with Separation of Variables

We show that Gammelgaard’s formula expressing a star product with separation of variables on a pseudo-Kähler manifold in terms of directed graphs without cycles is equivalent to an inversion formula for an operator on a formal Fock space. We prove this inversion formula directly and thus offer an alternative approach to Gammelgaard’s formula which gives more insight into the question why the directed graphs in his formula have no cycles.     

$${\mathcal{N}=2}$$ Superconformal Algebra and the Entropy of Calabi–Yau Manifolds

We use the representation theory of \({\mathcal{N}=2}\) superconformal algebra to study the elliptic genera of Calabi–Yau (CY) D-folds. We compute the entropy of CY manifolds from the growth rate of multiplicities of the massive (non-BPS) representations in the decomposition of their elliptic genera. We find that the entropy of CY manifolds of complex dimension D behaves differently depending on whether D is even or odd. When D is odd, CY entropy coincides with the entropy of the corresponding hyperKähler (D ? 3)-folds due to a structural theorem on Jacobi forms. In particular, we find that the Calabi–Yau 3-fold has a vanishing entropy. At D > 3, using our previous results on hyperKähler manifolds, we find \({S_{CY_D}\sim 2\pi \sqrt{\frac{(D-3)^2}{2(D-1)}n}}\). When D is even, we find the behavior of CY entropy behaving as \({S_{CY_D}\sim 2 \pi\sqrt{\frac{D-1}{2}n}}\). These agree with Cardy’s formula at large D.     

Inclusion relations for the spaces L(ℝ), \mathcal{H}\mathcal{K} (ℝ) ∩ \mathcal{B}\mathcal{V} (ℝ), and L 2(ℝ)

The inclusion relations for the spaces $ \mathcal{H}\mathcal{K} $ (I), L(I), $ \mathcal{H}\mathcal{K} $ (I) ∩ $ \mathcal{B}\mathcal{V} $ (I), and L ²(I) are found. On unbounded intervals, functions in $ \mathcal{H}\mathcal{K} $ (I) ∩ $ \mathcal{B}\mathcal{V} $ (I) need not be Lebesgue integrable.     

On Ruelle’s Construction of the Thermodynamic Limit for the Classical Microcanonical Entropy

In 1969 Ruelle published his construction of the thermodynamic limit, in the sense of Fisher, for the quasi-microcanonical entropy density of classical Hamiltonian N-body systems with stable and tempered pair interactions. Here, “quasi-microcanonical” refers to the fact that he discussed the entropy defined with a regularized microcanonical measure as ln (N!⁻¹ ∫ χ _{{ℰ−▵ℰ<H<ℰ}}d^6N X) rather than defined with the proper microcanonical measure as ln (N!⁻¹ ∫ δ(ℰ−H) d^6N X). Replacing δ(ℰ−H) by χ _{{ℰ−▵ℰ<H<ℰ}} seems to have become the standard procedure for rigorous treatments of the microcanonical ensemble hence. In this note we make a very elementary technical observation to the effect that Ruelle’s proof (still based on regularization) does establish the thermodynamic limit also for the entropy density defined with the proper microcanonical measure. We also show that with only minor changes in the proof the regularization of δ(ℰ−H) is actually not needed at all.