Related Experiment Video
Updated: Nov 27, 2025
06:42
Generation and Coherent Control of Pulsed Quantum Frequency Combs
Published on: June 8, 2018
9.4K
Unification of the Nature's Complexities via a Matrix Permanent-Critical Phenomena, Fractals, Quantum Computing,
Vitaly Kocharovsky1,2, Vladimir Kocharovsky2, Sergey Tarasov2
1Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843-4242, USA.
Entropy (Basel, Switzerland)
|December 8, 2020
Summary
This study reveals analytic connections between the matrix permanent and complex phenomena like critical phenomena, chaos, and quantum information. These insights stem from Ising model reductions and novel integral representations of the permanent.
Area of Science:
- Computational complexity
- Quantum information theory
- Statistical mechanics
- Number theory
- Fractal geometry
Background:
- The matrix permanent is a computationally challenging mathematical object.
- Its connections to diverse scientific fields are not fully understood.
- Understanding these connections can illuminate complex systems.
Purpose of the Study:
- To establish analytic relationships between the matrix permanent and various complex natural phenomena.
- To explore the permanent's role in critical phenomena, fractal structures, chaos, quantum information, and computational complexity.
Main Methods:
- Reduction of the Ising model (critical phenomena) to the matrix permanent.
- Development of four integral representations for the permanent.
- Utilizing fractal Weierstrass-like functions, complex variable polynomials, Laplace integrals, and the MacMahon master theorem.
Main Results:
- Demonstrated analytic relations between the matrix permanent and phenomena in physics and mathematics.
- Established the permanent as a unifying concept linking diverse complex systems.
- Provided novel integral representations offering new computational and theoretical avenues.
Conclusions:
- The matrix permanent serves as a fundamental link between seemingly disparate areas of science and mathematics.
- These findings offer new perspectives on complexity in critical phenomena, quantum systems, and computational problems.
- The developed integral representations are valuable tools for further research.
Related Concept Videos
01:20Parallel Processing
450
The brain processes sensory information rapidly due to parallel processing, which involves sending data across multiple neural pathways at the same time. This method allows the brain to manage various sensory qualities, such as shapes, colors, movements, and locations, all concurrently. For instance, when observing a forest landscape, the brain simultaneously processes the movement of leaves, the shapes of trees, the depth between them, and the various shades of green. This enables a quick and...
450
01:30Fundamental Theorem of Algebra
70
The Fundamental Theorem of Algebra is central to the study of polynomial equations, asserting that every non-constant polynomial with complex coefficients has at least one complex zero. This means that a polynomial of degree n ≥ 1, written as: with an ≠ 0, has at least one solution in the complex number system. Since the set of real numbers is a subset of complex numbers, this theorem applies equally to polynomials with real coefficients.Building on this result, the...
70
01:17Ampere-Maxwell's Law: Problem-Solving
938
A parallel-plate capacitor with capacitance C, whose plates have area A and separation distance d, is connected to a resistor R and a battery of voltage V. The current starts to flow at t = 0. What is the displacement current between the capacitor plates at time t? From the properties of the capacitor, what is the corresponding real current?
To solve the problem, we can use the equations from the analysis of an RC circuit and Maxwell's version of Ampère's law.
For the first part of the...
To solve the problem, we can use the equations from the analysis of an RC circuit and Maxwell's version of Ampère's law.
For the first part of the...
938
02:12The Wave Nature of Light
58.9K
The nature of light has been a subject of inquiry since antiquity. In the seventeenth century, Isaac Newton performed experiments with lenses and prisms and was able to demonstrate that white light consists of the individual colors of the rainbow combined together. Newton explained his optics findings in terms of a "corpuscular" view of light, in which light was composed of streams of extremely tiny particles traveling at high speeds according to Newton's laws of motion.
58.9K
01:25Multimachine Stability
308
Multimachine stability analysis is crucial for understanding the dynamics and stability of power systems with multiple synchronous machines. The objective is to solve the swing equations for a network of M machines connected to an N-bus power system.
In analyzing the system, the nodal equations represent the relationship between bus voltages, machine voltages, and machine currents. The nodal equation is given by:
In analyzing the system, the nodal equations represent the relationship between bus voltages, machine voltages, and machine currents. The nodal equation is given by:
308
01:19Cyclic Processes And Isolated Systems
3.2K
A thermodynamic system with zero heat exchange and work is an isolated system. For these systems, the internal energy remains constant.
In the case of a non-isolated system, the change in the internal energy is zero only if the process is cyclic. A thermodynamic process is considered cyclic if the system undergoes a series of changes and returns to its initial state.
Consider a cyclic process that returns to its initial state, undergoing a four-step process. The heat transfer along each...
In the case of a non-isolated system, the change in the internal energy is zero only if the process is cyclic. A thermodynamic process is considered cyclic if the system undergoes a series of changes and returns to its initial state.
Consider a cyclic process that returns to its initial state, undergoing a four-step process. The heat transfer along each...
3.2K