The Michaelis-Menten-Stueckelberg Theorem

We study chemical reactions with complex mechanisms under two assumptions: (i) intermediates are present in small amounts (this is the quasi-steady-state hypothesis or QSS) and (ii) they are in equilibrium relations with substrates (this is the quasiequilibrium hypothesis or QE). Under these assumptions, we prove the generalized mass action law together with the basic relations between kinetic factors, which are sufficient for the positivity of the entropy production but hold even without microreversibility, when the detailed balance is not applicable. Even though QE and QSS produce useful approximations by themselves, only the combination of these assumptions can render the possibility beyond the "rarefied gas" limit or the "molecular chaos" hypotheses. We do not use any a priori form of the kinetic law for the chemical reactions and describe their equilibria by thermodynamic relations. The transformations of the intermediate compounds can be described by the Markov kinetics because of their low density ({\em low density of elementary events}). This combination of assumptions was introduced by Michaelis and Menten in 1913. In 1952, Stueckelberg used the same assumptions for the gas kinetics and produced the remarkable semi-detailed balance relations between collision rates in the Boltzmann equation that are weaker than the detailed balance conditions but are still sufficient for the Boltzmann $H$-theorem to be valid. Our results are obtained within the Michaelis-Menten-Stueckelbeg conceptual framework.Comment: 54 pages, the final version; correction of a misprint in Attachment

Reciprocal Relations Between Kinetic Curves

We study coupled irreversible processes. For linear or linearized kinetics with microreversibility, $\dot{x}=Kx$, the kinetic operator $K$ is symmetric in the entropic inner product. This form of Onsager's reciprocal relations implies that the shift in time, $\exp (Kt)$, is also a symmetric operator. This generates the reciprocity relations between the kinetic curves. For example, for the Master equation, if we start the process from the $i$th pure state and measure the probability $p_j(t)$ of the $j$th state ($j\neq i$), and, similarly, measure $p_i(t)$ for the process, which starts at the $j$th pure state, then the ratio of these two probabilities $p_j(t)/p_i(t)$ is constant in time and coincides with the ratio of the equilibrium probabilities. We study similar and more general reciprocal relations between the kinetic curves. The experimental evidence provided as an example is from the reversible water gas shift reaction over iron oxide catalyst. The experimental data are obtained using Temporal Analysis of Products (TAP) pulse-response studies. These offer excellent confirmation within the experimental error.Comment: 6 pages, 1 figure, the final versio

Infrared Yang-Mills theory as a spin system. A lattice approach

To verify the conjecture that Yang-Mills theory in the infrared limit is equivalent to a spin system whose excitations are knot solitons, a numerical algorithm based on the inverse Monte Carlo method is proposed. To investigate the stability of the effective spin field action, numerical studies of the renormalization group flow for the coupling constants are suggested. A universality of the effective spin field action is also discussed.Comment: Latex 12 pages, no figures, references added, some comments added, to appear in Phys.Lett.

Moment bounds for SPDEs with non-Gaussian fields and application to the Wong-Zakai problem

Upon its inception the theory of regularity structures allowed for the treatment for many semilinear perturbations of the stochastic heat equation driven by space-time white noise. When the driving noise is non-Gaussian the machinery of theory can still be used but must be combined with an infinite number of stochastic estimates in order to compensate for the loss of hypercontractivity. In this paper we obtain a more streamlined and automatic set of criteria implying these estimates which facilitates the treatment of some other problems including non-Gaussian noise such as some general phase coexistence models - as an example we prove here a generalization of the Wong-Zakai Theorem found by Hairer and Pardoux.Comment: 37 page

Quantum vs Classical Integrability in Ruijsenaars-Schneider Systems

The relationship (resemblance and/or contrast) between quantum and classical integrability in Ruijsenaars-Schneider systems, which are one parameter deformation of Calogero-Moser systems, is addressed. Many remarkable properties of classical Calogero and Sutherland systems (based on any root system) at equilibrium are reported in a previous paper (Corrigan-Sasaki). For example, the minimum energies, frequencies of small oscillations and the eigenvalues of Lax pair matrices at equilibrium are all "integer valued". In this paper we report that similar features and results hold for the Ruijsenaars-Schneider type of integrable systems based on the classical root systems.Comment: LaTeX2e with amsfonts 15 pages, no figure

Non-canonical extension of theta-functions and modular integrability of theta-constants

This is an extended (factor 2.5) version of arXiv:math/0601371 and arXiv:0808.3486. We present new results in the theory of the classical $\theta$-functions of Jacobi: series expansions and defining ordinary differential equations (\odes). The proposed dynamical systems turn out to be Hamiltonian and define fundamental differential properties of theta-functions; they also yield an exponential quadratic extension of the canonical $\theta$-series. An integrability condition of these \odes\ explains appearance of the modular $\vartheta$-constants and differential properties thereof. General solutions to all the \odes\ are given. For completeness, we also solve the Weierstrassian elliptic modular inversion problem and consider its consequences. As a nontrivial application, we apply proposed techni\-que to the Hitchin case of the sixth Painlev\'e equation.Comment: Final version; 47 pages, 1 figure, LaTe