scholarly journals Statistical Mechanics Involving Fractal Temperature

2019 ◽  
Vol 3 (2) ◽  
pp. 20 ◽  
Author(s):  
Alireza Khalili Golmankhaneh
Keyword(s):  
Heat Capacity ◽  
Partition Function ◽  
Statistical Mechanics ◽  
Density Of States ◽  
Schrodinger Equation ◽  
Time Derivative ◽  
Fractional Dimension ◽  
Eigenvalues And Eigenfunctions ◽  
Solid Models ◽  
Fractal Time

In this paper, the Schrödinger equation involving a fractal time derivative is solved and corresponding eigenvalues and eigenfunctions are given. A partition function for fractal eigenvalues is defined. For generalizing thermodynamics, fractal temperature is considered, and adapted equations are defined. As an application, we present fractal Dulong-Petit, Debye, and Einstein solid models and corresponding fractal heat capacity. Furthermore, the density of states for fractal spaces with fractional dimension is obtained. Graphs and examples are given to show details.

Theory of Unimolecular Decomposition— The Statistical Approach

1996 ◽  
Author(s):  
Tomas Baer ◽  
William L. Hase
Keyword(s):  
Partition Function ◽  
Statistical Mechanics ◽  
Density Of States ◽  
Quantum Statistical Mechanics ◽  
Thermodynamic Quantities ◽  
Unimolecular Decomposition ◽  
Thermal Systems ◽  
System P ◽  
Quantum Statistical ◽  
Basic Ideas

The partition function and the sum or density of states are functions which are to statistical mechanics what the wave function is to quantum mechanics. Once they are known, all of the thermodynamic quantities of interest can be calculated. It is instructive to compare these two functions because they are closely related. Both provide a measure of the number of states in a system. The partition function is a quantity that is appropriate for thermal systems at a given temperature (canonical ensemble), whereas the sum and density of states are equivalent functions for systems at constant energy (microcanonical ensemble). In order to lay the groundwork for an understanding of these two functions as well as a number of other topics in the theory of unimolecular reactions, it is essential to review some basic ideas from classical and quantum statistical mechanics. As discussed in chapter 2, the classical Hamiltonian, H(p,q), is the total energy of the system expressed in terms of the momenta (p) and positions (q) of the atoms in the system.

On the quantum mechanical solutions with minimal length uncertainty

2016 ◽  
Vol 31 (18) ◽  
pp. 1650101 ◽  
Author(s):  
Homa Shababi ◽  
Pouria Pedram ◽  
Won Sang Chung
Keyword(s):  
Heat Capacity ◽  
Partition Function ◽  
Harmonic Oscillator ◽  
Internal Energy ◽  
Free Particle ◽  
Minimal Length ◽  
Quantum Mechanical ◽  
Uncertainty Principles ◽  
Trigonometric Functions ◽  
Eigenvalues And Eigenfunctions

In this paper, we study two generalized uncertainty principles (GUPs) including [Formula: see text] and [Formula: see text] which imply minimal measurable lengths. Using two momentum representations, for the former GUP, we find eigenvalues and eigenfunctions of the free particle and the harmonic oscillator in terms of generalized trigonometric functions. Also, for the latter GUP, we obtain quantum mechanical solutions of a particle in a box and harmonic oscillator. Finally we investigate the statistical properties of the harmonic oscillator including partition function, internal energy, and heat capacity in the context of the first GUP.

scholarly journals General Quantum Theory: I ---- Deducing Quantum Physics and Solving Two Origin Crisises of Wave-Particle Duality and the First Quantization

2021 ◽  
Author(s):  
C. Huang ◽  
Yong-Chang Huang ◽  
Jia-Min Song
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Statistical Mechanics ◽  
Schrodinger Equation ◽  
Quantum Physics ◽  
Time Derivative ◽  
Square Root ◽  
Classical Statistical Mechanics ◽  
Wave Particle Duality ◽  
Solid Bases

Density distribution function of classical statistical mechanics is generally generalized as a product of a general complex function and its complex Hermitian conjugate function, and the average of classical statistical mechanics is generalized as the average of the quantum mechanics. Furthermore, this paper derives three ones of the five axiom presumptions of quantum mechanics, e.g., deduces Schrȍdinger equation by two general ways, makes the three axiom presumptions into three theorems of quantum mechanics, not only solves the crisis to hard understand, but also gets new theories and new discoveries, e.g., this paper solves the crisis of the origin of the wave-particle duality, derives operators, eigenvalues and eigenstates, deduces commutation relations for coordinate and momentum as well as the time and energy, and discovers quantum mechanics is just a generalization ( mechanics ) theory of the complex square root of ( real density function of ) classical statistical mechanics. Quantum mechanics being just a generalization theory of the complex square root of classical statistical mechanics is both new physics and revolutionary discovery, which are affecting people’s deep philosophical thinking for modern physics development, solve all the crisises of quantum mechanics, quantum information and so on, and make quantum mechanics have scientific solid bases being checked and both no basic axiom presumption and no all the quantum strange incomprehensible properties, because classical statistical mechanics and the complex square root of classical statistical mechanics have the scientific solid bases being checked. In addition, this paper discovers the reason no taking the time derivative of space coordinates in Schrȍdinger equation. Therefore, this paper gives solution to the crisis of the first quantization origin, and mainly deduces quantum physics no all the quantum current strange incomprehensible properties.

scholarly journals General Quantum Theory No Axiom Presumption: I ----Quantum Mechanics and Solutions to Crisises of Origins of Both Wave-Particle Duality and the First Quantization

2020 ◽  
Author(s):  
C. Huang ◽  
Yong-Chang Huang ◽  
Jia-Min Song
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Statistical Mechanics ◽  
Schrodinger Equation ◽  
Time Derivative ◽  
Complex Function ◽  
Conjugate Function ◽  
Classical Statistical Mechanics ◽  
Wave Particle Duality ◽  
Modern Physics

Density distribution function of classical statistical mechanics is generally generalized as a product of a general complex function and its complex Hermitian conjugate function, and the average of classical statistical mechanics is generalized as the average of the quantum mechanics. Furthermore, this paper derives three ones of the five axiom presumptions of quantum mechanics, e.g., deduces Schrȍdinger equation by two general ways, makes the three axiom presumptions into three theorems of quantum mechanics, not only solves the crisis to hard understand, but also gets new theories and new discoveries, e.g., this paper solves the crisis of the origin of the wave-particle duality (i.e., complementary principle), derives operators, eigenvalues and eigenstates, deduces commutation relations for coordinate and momentum as well as the time and energy, and discovers quantum mechanics is just a generalization ( mechanics ) theory of the complex square root of ( real density function of ) classical statistical mechanics, which will make people renew thinking modern physics development. In addition, this paper discovers the reason that Schrȍdinger equation doesn’t takes the time derivative of space coordinates. Therefore, this paper gives solution to the Crisis of the first quantization origin.

scholarly journals Hard Squares with Negative Activity and Rhombus Tilings of the Plane

10.37236/1093 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Jakob Jonsson
Keyword(s):  
Partition Function ◽  
Statistical Mechanics ◽  
Simplicial Complex ◽  
Transfer Matrix ◽  
Euler Characteristic ◽  
Independent Sets ◽  
Roots Of Unity ◽  
Hard Squares ◽  
Vertex Set

Let $S_{m,n}$ be the graph on the vertex set ${\Bbb Z}_m \times {\Bbb Z}_n$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,d\pm 1)$ or $(a,b) = (c \pm 1,d)$ modulo $(m,n)$. We present a formula for the Euler characteristic of the simplicial complex $\Sigma_{m,n}$ of independent sets in $S_{m,n}$. In particular, we show that the unreduced Euler characteristic of $\Sigma_{m,n}$ vanishes whenever $m$ and $n$ are coprime, thereby settling a conjecture in statistical mechanics due to Fendley, Schoutens and van Eerten. For general $m$ and $n$, we relate the Euler characteristic of $\Sigma_{m,n}$ to certain periodic rhombus tilings of the plane. Using this correspondence, we settle another conjecture due to Fendley et al., which states that all roots of $\det (xI-T_m)$ are roots of unity, where $T_m$ is a certain transfer matrix associated to $\{\Sigma_{m,n} : n \ge 1\}$. In the language of statistical mechanics, the reduced Euler characteristic of $\Sigma_{m,n}$ coincides with minus the partition function of the corresponding hard square model with activity $-1$.

scholarly journals Statistical Mechanics of Discrete Multicomponent Fragmentation

2020 ◽  
Vol 5 (4) ◽  
pp. 64
Author(s):  
Themis Matsoukas
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Partition Function ◽  
Statistical Mechanics ◽  
Closed Form ◽  
Fixed Number ◽  
Arbitrary Degree ◽  
Fragment Distribution ◽  
The Mean ◽  
Fragmentation Event

We formulate the statistics of the discrete multicomponent fragmentation event using a methodology borrowed from statistical mechanics. We generate the ensemble of all feasible distributions that can be formed when a single integer multicomponent mass is broken into fixed number of fragments and calculate the combinatorial multiplicity of all distributions in the set. We define random fragmentation by the condition that the probability of distribution be proportional to its multiplicity, and obtain the partition function and the mean distribution in closed form. We then introduce a functional that biases the probability of distribution to produce in a systematic manner fragment distributions that deviate to any arbitrary degree from the random case. We corroborate the results of the theory by Monte Carlo simulation, and demonstrate examples in which components in sieve cuts of the fragment distribution undergo preferential mixing or segregation relative to the parent particle.

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