Related papers: Unbounded Probability Theory and Its Applications
This expository paper advocates an approach to physics in which ``typicality" is identified with a suitable form of algorithmic randomness. To this end various theorems from mathematics and physics are reviewed. Their original versions…
We present novel bounds for estimating discrete probability distributions under the $\ell_\infty$ norm. These are nearly optimal in various precise senses, including a kind of instance-optimality. Our data-dependent convergence guarantees…
Complex continuous or mixed joint distributions (e.g., P(Y | z_1, z_2, ..., z_N)) generally lack closed-form solutions, often necessitating approximations such as MCMC. This paper proposes Indeterminate Probability Theory (IPT), which makes…
In the real world, one almost never knows the parameters of a thermodynamic process to infinite precision. Reflecting this, here we investigate how to extend stochastic thermodynamics to systems with uncertain parameters, including…
Turbulent dynamical systems characterized by both a high-dimensional phase space and a large number of instabilities are ubiquitous among many complex systems in science and engineering. The existence of a strange attractor in the turbulent…
We review some recent developments which make use of the concept of `superstatistics', an effective description for nonequilibrium systems with a varying intensive parameter such as the inverse temperature. We describe how the asymptotic…
Statistical thermodynamics has a universal appeal that extends beyond molecular systems, and yet, as its tools are being transplanted to fields outside physics, the fundamental question, \textit{what is thermodynamics?}, has remained…
Some inequalities for probability vector are discussed. The probability representation of quantum mechanics where the states are mapped onto probability vectors (either finite or infinite dimensional) called the state tomograms is used.…
Despite its enormous empirical success, the formalism of quantum theory still raises fundamental questions: why is nature described in terms of complex Hilbert spaces, and what modifications of it could we reasonably expect to find in some…
Algorithmic theories of randomness can be related to theories of probabilistic sequence prediction through the notion of a predictor, defined as a function which supplies lower bounds on initial-segment probabilities of infinite sequences.…
We formulate thermodynamics of economic systems in terms of an arbitrary probability distribution for a conserved economic quantity. As in statistical physics, thermodynamic macroeconomic variables emerge as the mean value of microeconomic…
We construct a class of nonnegative martingale processes that oscillate indefinitely with high probability. For these processes, we state a uniform rate of the number of oscillations and show that this rate is asymptotically close to the…
We use Lyapunov-like functions and convex optimization to propagate uncertainty in the initial condition of nonlinear systems governed by ordinary differential equations. We consider the full nonlinear dynamics without approximation,…
Game-theoretic upper expectations are joint (global) probability models that mathematically describe the behaviour of uncertain processes in terms of supermartingales; capital processes corresponding to available betting strategies.…
The domain of validity of standard thermodynamics and Boltzmann-Gibbs statistical mechanics is discussed and then formally enlarged in order to hopefully cover a variety of anomalous systems. The generalization concerns {\it nonextensive}…
Bohr and Heisenberg suggested that the thermodynamical quantities of temperature and energy are complementary in the same way as position and momentum in quantum mechanics. Roughly speaking, their idea was that a definite temperature can be…
We develop an analogue of probability theory for probabilities taking values in topological groups. We generalize Kolmogorov's method of axiomatization of probability theory: main distinguishing features of frequency probabilities are taken…
We present the full thermodynamics of a fluid confined by an arbitrary external potential based on the virial expansion of the grand potential. The fluid may be classical or quantum and it is assumed that interatomic interactions are…
The Lorentz covariant statistical physics and thermodynamics is formulated within the preferred frame approach. The transformation laws for geometrical and mechanical quantities such as volume and pressure as well as the Lorentz-invariant…
Small, out-of-equilibrium, and quantum systems defy simple thermodynamic expressions. Such systems are exemplified by molecular switches, which exchange heat with a bath. These molecules can photoisomerize, or change conformation, or…