scholarly journals Algorithmic thermodynamics

2012 ◽  
Vol 22 (5) ◽  
pp. 771-787 ◽  
Author(s):  
JOHN BAEZ ◽  
MIKE STAY
Keyword(s):  
Partition Function ◽  
Chemical Potential ◽  
Algorithmic Information Theory ◽  
Thermodynamic Relation ◽  
Thermodynamic Cycles ◽  
Heat Engines ◽  
Algorithmic Information ◽  
The Subject ◽  
Expected Values ◽  
Special Case

Algorithmic entropy can be viewed as a special case of the entropy studied in statistical mechanics. This viewpoint allows us to apply many techniques developed for use in thermodynamics to the subject of algorithmic information theory. In particular, suppose we fix a universal prefix-free Turing machine and let X be the set of programs that halt for this machine. Then we can regard X as a set of ‘microstates’, and treat any function on X as an ‘observable’. For any collection of observables, we can study the Gibbs ensemble that maximises entropy subject to constraints on the expected values of these observables. We illustrate this by taking the log runtime, length and output of a program as observables analogous to the energy E, volume V and number of molecules N in a container of gas. The conjugate variables of these observables allow us to define quantities we call the ‘algorithmic temperature’ T, ‘algorithmic pressure’ P and ‘algorithmic potential’ μ, since they are analogous to the temperature, pressure and chemical potential. We derive an analogue of the fundamental thermodynamic relation dE = TdS − PdV + μdN, and use it to study thermodynamic cycles analogous to those for heat engines. We also investigate the values of T, P and μ for which the partition function converges. At some points on the boundary of this domain of convergence, the partition function becomes uncomputable – indeed, at these points the partition function itself has non-trivial algorithmic entropy.

scholarly journals Modular invariance in finite temperature Casimir effect

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Francesco Alessio ◽  
Glenn Barnich
Keyword(s):  
Partition Function ◽  
Chemical Potential ◽  
Casimir Effect ◽  
Temperature Inversion ◽  
Massless Scalar ◽  
Partition Functions ◽  
Parallel Plates ◽  
Modular Transformations ◽  
Real Analytic ◽  
Set Up

Abstract The temperature inversion symmetry of the partition function of the electromagnetic field in the set-up of the Casimir effect is extended to full modular transformations by turning on a purely imaginary chemical potential for adapted spin angular momentum. The extended partition function is expressed in terms of a real analytic Eisenstein series. These results become transparent after explicitly showing equivalence of the partition functions for Maxwell’s theory between perfectly conducting parallel plates and for a massless scalar with periodic boundary conditions.

Kinematic Investigation of the Deployable Bennett Loop

2006 ◽  
Vol 129 (6) ◽  
pp. 602-610 ◽  
Author(s):  
J. Eddie Baker
Keyword(s):  
Full Advantage ◽  
Spatial Displacement ◽  
Learning Tool ◽  
Working Mechanism ◽  
Efficient Treatment ◽  
Kinematic Characteristics ◽  
The Subject ◽  
The Many ◽  
Special Case

Despite the many studies devoted to it and its value as a learning tool, the Bennett linkage has never been employed as a working mechanism. It has recently found favor, however, among structural analysts as a possible unit in deployable networks owing to the potential for true spatial displacement without flexure. Although the loop can be analyzed in this application by means of purely geometrical methods, a wealth of kinematic examinations is available for more efficient treatment. The particular form that the chain must adopt as a deployable object and the special case of the linkage demanded by the purpose constitute the subject of the present exposition, which takes full advantage of prior analyses of the chain’s kinematic characteristics.

1. On Cases of Instability in Open Structures

1888 ◽  
Vol 14 ◽  
pp. 106-106
Author(s):  
E. Sang
Keyword(s):  
Linear Structures ◽  
Symmetric Structure ◽  
Linear Resistance ◽  
Open Structures ◽  
Extensive Class ◽  
The Subject ◽  
The Stability ◽  
Special Case

AbstractIn the course of some remarks on the design proposed for the Forth Bridge, the author of this paper had enunciated the remarkable theorem, that any symmetric structure built on a rectangular base, and depending on linear resistance alone, is necessarily unstable. The proof of it, given in the eleventh volume of the Transactions of the Royal Scottish Society of Arts, is derived from considerations affecting the special case; but this theorem is only one of an extensive class, and therefore the subject of instability among linear structures in general is here taken up.In the case of regular or semi-regular arrangements, having the corners of an upper supported from the corners of an under polygon, it is shown that when the figures are of odd numbers the structures are stable, while those with even numbers are unstable ; unless indeed the polygons be placed conformably, in which case the stability extends to both classes.

scholarly journals Notes on massless scalar field partition functions, modular invariance and Eisenstein series

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Francesco Alessio ◽  
Glenn Barnich ◽  
Martin Bonte
Keyword(s):  
Scalar Field ◽  
Partition Function ◽  
Low Temperature ◽  
Eisenstein Series ◽  
Chemical Potential ◽  
Proper Time ◽  
Series Representation ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Spacetime Manifold

Abstract The partition function of a massless scalar field on a Euclidean spacetime manifold ℝd−1 × 𝕋2 and with momentum operator in the compact spatial dimension coupled through a purely imaginary chemical potential is computed. It is modular covariant and admits a simple expression in terms of a real analytic SL(2, ℤ) Eisenstein series with s = (d + 1)/2. Different techniques for computing the partition function illustrate complementary aspects of the Eisenstein series: the functional approach gives its series representation, the operator approach yields its Fourier series, while the proper time/heat kernel/world-line approach shows that it is the Mellin transform of a Riemann theta function. High/low temperature duality is generalized to the case of a non-vanishing chemical potential. By clarifying the dependence of the partition function on the geometry of the torus, we discuss how modular covariance is a consequence of full SL(2, ℤ) invariance. When the spacetime manifold is ℝp × 𝕋q+1, the partition function is given in terms of a SL(q + 1, ℤ) Eisenstein series again with s = (d + 1)/2. In this case, we obtain the high/low temperature duality through a suitably adapted dual parametrization of the lattice defining the torus. On 𝕋d+1, the computation is more subtle. An additional divergence leads to an harmonic anomaly.

scholarly journals Independence Conservation and Evolutionary Algorithms

2020 ◽  
Vol 2 (1) ◽  
pp. 32-35
Author(s):  
Eric Holloway
Keyword(s):  
Information Theory ◽  
Evolutionary Algorithms ◽  
Algorithmic Information Theory ◽  
Fitness Functions ◽  
Information Law ◽  
Algorithmic Information

Leonid Levin developed the first stochastic conservation of information law, describing it as "torturing an uninformed witness cannot give information about the crime."  Levin's law unifies both the deterministic and stochastic cases of conservation of information.  A proof of Levin's law from Algorithmic Information Theory is given as well as a discussion of its implications in evolutionary algorithms and fitness functions.

scholarly journals From Information and Quantum Physics to Consciousness and Reality

Sci ◽  
2021 ◽  
Vol 3 (4) ◽  
pp. 35
Author(s):  
Peter Verheyen
Keyword(s):  
Quantum Physics ◽  
Entanglement Entropy ◽  
Relativity Theory ◽  
Recent Theory ◽  
Algorithmic Information Theory ◽  
Person Perspective ◽  
The World ◽  
Algorithmic Information ◽  
Role Of Information

How does the world around us work and what is real? This question has preoccupied humanity since its beginnings. From the 16th century onwards, it has periodically been necessary to revise the prevailing worldview—but things became very strange at the beginning of the 20th century with the advent of relativity theory and quantum physics. The current focus is on the role of information, there being a debate about whether this is ontological or epistemological. A theory has recently been formulated in which spacetime and gravity emerges from microscopic quantum information—more specifically from quantum entanglement via entanglement entropy. A recent theory describes the emergence of reality itself through first-person perspective experiences and algorithmic information theory. In quantum physics, perception and observation play a central role. Perception of and interaction with the environment require an exchange of information. Via biochemical projection, information is given an interpretation that is necessary to make life and consciousness possible. The world around us is not at all what it seems.

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