Related Experiment Video
Updated: Jan 12, 2026
11:03
An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
Published on: December 4, 2017
9.0K
Universality of the Microcanonical Entropy at Large Spin
Sridip Pal1, Jiaxin Qiao2, Balt C van Rees3
1Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA USA.
Communications in Mathematical Physics
|November 3, 2025
Summary
This study reveals how modular invariance impacts two-dimensional conformal field theories (CFTs). The spectral density of spin-J operators grows exponentially with J, indicating a dense spectrum in CFTs with c > 1.
Area of Science:
- Theoretical Physics
- Quantum Field Theory
- String Theory
Background:
- Modular invariance is a key principle in 2D CFTs.
- Non-rational CFTs with c > 1 exhibit complex spectral properties.
- Understanding operator spectra is crucial for CFT analysis.
Purpose of the Study:
- To rigorously investigate the consequences of modular invariance in 2D non-rational CFTs.
- To determine the growth rate of the spectral density of spin-J operators.
- To analyze the behavior of operator spectra at different twist intervals.
Main Methods:
- Analysis of the torus partition function.
- Derivation of spectral density estimates using modular invariance.
- Asymptotic analysis of operator spin and twist.
Main Results:
- The spectral density of spin-J operators grows as exp(π√(2(c-1)J/3))/√(2J) for twists ≥ (c-1)/12.
- This growth rate proves spectral density becomes dense for large J, even without spin averaging.
- For twists < (c-1)/12, the spectral density growth is strictly slower.
Conclusions:
- Modular invariance imposes strong constraints on operator spectra in 2D CFTs.
- The derived spectral density growth confirms the dense nature of spectra in non-rational CFTs.
- The study provides estimates for the maximal gap between spin-J operators.
Related Concept Videos
02:39Entropy
34.8K
Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...
34.8K
01:18Entropy
3.5K
The first law of thermodynamics is quantitatively formulated via an equation relating the internal energy of a system, the heat exchanged by it, and the work done on it. A quantitative formulation of the second law of thermodynamics leads to defining a state function, the entropy.
When an ideal gas expands isothermally, the disorder in the gas increases. From the molecular perspective, the gas molecules have more volume to move around in.
Consider an infinitesimal step in the expansion, which...
When an ideal gas expands isothermally, the disorder in the gas increases. From the molecular perspective, the gas molecules have more volume to move around in.
Consider an infinitesimal step in the expansion, which...
3.5K
02:38Third Law of Thermodynamics
21.5K
A pure, perfectly crystalline solid possessing no kinetic energy (that is, at a temperature of absolute zero, 0 K) may be described by a single microstate, as its purity, perfect crystallinity,and complete lack of motion means there is but one possible location for each identical atom or molecule comprising the crystal (W = 1). According to the Boltzmann equation, the entropy of this system is zero.
21.5K
01:20Entropy and the Second Law of Thermodynamics
4.7K
The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
The relation between entropy and disorder can be illustrated with the example of the phase change of ice to water. In ice, the molecules are located at specific sites giving a solid state, whereas, in a liquid form, these molecules are much freer to move. The molecular arrangement has therefore become more randomized. Although the change in average...
The relation between entropy and disorder can be illustrated with the example of the phase change of ice to water. In ice, the molecules are located at specific sites giving a solid state, whereas, in a liquid form, these molecules are much freer to move. The molecular arrangement has therefore become more randomized. Although the change in average...
4.7K
01:14Atomic Nuclei: Nuclear Spin State Population Distribution
2.3K
Near absolute zero temperatures, in the presence of a magnetic field, the majority of nuclei prefer the lower energy spin-up state to the higher energy spin-down state. As temperatures increase, the energy from thermal collisions distributes the spins more equally between the two states. The Boltzmann distribution equation gives the ratio of the number of spins predicted in the spin −½ (N−) and spin +½ (N+) states.
2.3K
01:14The Second Law of Thermodynamics
6.6K
In the quest to identify a property that may reliably predict the spontaneity of a process, a promising candidate has been identified: entropy. Scientists refer to the measure of randomness or disorder within a system as entropy. High entropy means high disorder and low energy. To better understand entropy, think of a student’s bedroom. If no energy or work were put into it, the room would quickly become messy. It would exist in a very disordered state, one of high entropy. Energy must be...
6.6K