A method and system for generating steady-state quantum turbulence with conservation of particle number

Abstract

This invention proposes a method and system for generating steady-state quantum turbulence while preserving particle number conservation. The method includes: constructing an evolutionary model of a quantum fluid system, which is obtained by introducing an external driving term and a particle number conservation dissipation term based on the Gross-Pitaevskii equation; the external driving term is a spatiotemporally varying external potential used to inject energy into the system; the particle number conservation dissipation term depends on the time-varying rate of particle number density and is used to dissipate the system's energy; both the external driving term and the particle number conservation dissipation term are configured to maintain the total particle number conservation of the system; and the quantum fluid system is subjected to time evolution based on the evolutionary model until the system reaches a steady-state quantum turbulence state with dynamic equilibrium of total kinetic energy. This invention eliminates the need to compensate for particle loss by adjusting the chemical potential, can stably generate and control quantum turbulence structures at different scales, and ensures the physical self-consistency and long-term stability of the quantum fluid evolution process.

Description

Technical Field

[0001] This invention belongs to the field of quantum fluids and relates to a method and system for generating steady-state quantum turbulence, and more particularly to a method and system for generating steady-state quantum turbulence while preserving particle number conservation. Background Technology

[0002] Quantum turbulence refers to complex nonequilibrium flow phenomena occurring in superfluids, Bose-Einstein condensates, and other quantum fluid systems, and is widely found in ultracold atom physics, nonlinear optics, and quantum fluid-like simulations. Compared to classical turbulence, quantum turbulence is simultaneously influenced by quantum coherence, nonlinear interactions, and dissipation processes; its formation and maintenance depend on the dynamic balance between driving forces and dissipation. Therefore, how to stably generate and maintain statistically steady-state quantum turbulence under controllable conditions has become a crucial technical problem in current quantum fluid dynamics research and related applications.

[0003] In existing technologies, the generation of quantum turbulence typically relies on numerical or experimental realization of nonlinear Schrödinger or Gross-Pitaevskii equations, triggering the formation of complex flow structures through instantaneous external field excitation, random perturbations, and other methods. To maintain the turbulent state, some schemes directly introduce dissipation mechanisms related to the amplitude or phase of the wave function into the evolution equations, or adjust the chemical potential to compensate for particle loss, thereby maintaining the average number of particles in the system over a certain timescale.

[0004] However, the aforementioned existing technologies generally suffer from the following shortcomings: First, directly introducing general dissipation terms often disrupts the continuity equations of the system, leading to non-conservation of the condensed particle number and making it difficult to define a steady state effectively. Second, methods that maintain the particle number by dynamically adjusting the chemical potential increase control complexity and are difficult to guarantee stability during long-term evolution. Third, the strong coupling between the driving and dissipation mechanisms makes it difficult to independently control the energy injection scale and dissipation process, limiting the systematic study of the structure and statistical properties of steady-state quantum turbulence. Therefore, there is an urgent need for a technical solution that can achieve continuous driving and stable maintenance of statistical steady-state quantum turbulence while maintaining particle number conservation. Summary of the Invention

[0005] To address the aforementioned problems, the present invention aims to provide a method and system for generating steady-state quantum turbulence while preserving particle number conservation. By combining a dissipative form that preserves particle number conservation with a potential-based driving method, quantum fluids can form long-term stable steady-state quantum turbulence.

[0006] The technical solution adopted in this invention is as follows:

[0007] A method for generating steady-state quantum turbulence while preserving particle number conservation includes the following steps:

[0008] An evolutionary model of a quantum fluid system is constructed, which is derived from the Gross-Pitaevskii equation by introducing an external driving term and a particle number conservation dissipation term; wherein:

[0009] The external driving term is an additional external potential that varies in space and time, used to inject energy into the system; the particle number conservation dissipation term depends on the time rate of change of particle number density, used to dissipate the energy of the system; both the external driving term and the particle number conservation dissipation term are configured to maintain the total particle number of the system.

[0010] The quantum fluid system is subjected to time evolution based on the aforementioned evolution model until the system reaches a steady-state quantum turbulence state with dynamic equilibrium of total kinetic energy.

[0011] Furthermore, the external driving term is introduced in the following manner:

[0012] Based on the fluid dynamics form of the Gross-Pitaevskii equations, the flow velocity is driven by introducing a driving force into the Euler equations while keeping the continuity equations unchanged.

[0013] The driving force is a conservative force, and the potential function corresponding to the conservative force is introduced into the Gross-Pitaevskii equation as the additional external potential.

[0014] Furthermore, by constructing a potential function, the driving force of spatiotemporal changes can be obtained, thereby controlling the driving scale and driving intensity. Specific steps include:

[0015] Define the range of the driving force's effective scale k in momentum space. V And construct a piecewise function V k :

[0016] ,

[0017] in This indicates how the driving force is generated in momentum space. A random phase within [0, 1] Where L is the wave number and L is the size of the spatial region;

[0018] For V k Performing the inverse Fourier transform yields the fundamental potential field V in coordinate space. x and the basic potential field V x The amplitude is normalized to [0, V0], where This represents the maximum driving force.

[0019] The potential function is updated using an iterative formula with time correlation:

[0020] ,

[0021] in, Let be the potential function at the current time t. is the potential function at the next time step after the update; T is the time variation period, used to control the rate of energy injection and the driving intensity; R is the correlation factor, used to control the degree of correlation between driving potentials at adjacent time steps;

[0022] The updated potential function The amplitude is renormalized to [0, V0].

[0023] Furthermore, the specific form of the particle number conservation dissipation term in the Gross–Pitaevskii equation is as follows:

[0024] ,

[0025] in, Let i be the superfluid condensation wave function, and i be an imaginary number. Let g be the chemical potential, g be the nonlinear interaction strength, and t be time. For spatial gradient operators; The particle number conservation dissipation term ensures that the total energy decay is only related to the time rate of change of particle number density, and λ is the dissipation coefficient used to control the dissipation intensity.

[0026] Furthermore, the particle number conservation dissipation term is constructed into a spatial derivative form using the continuity equation, and the Gross–Pitaevskii equation with the particle number conservation dissipation term is expressed as:

[0027] ,

[0028] in, This indicates complex conjugation, and Im[] represents the imaginary part extraction operation.

[0029] Furthermore, the evolutionary model is composed of the aforementioned external driving term and the aforementioned particle number conservation dissipation term.

[0030] Furthermore, the evolutionary model is expressed as follows:

[0031] ,

[0032] in, Let i be the superfluid condensation wave function, and i be an imaginary number. Where is the chemical potential, g is the nonlinear interaction strength, and V(t,x) is the external driving term. For evolution time, In real space coordinates; Here, λ is the dissipation term for particle number conservation, and λ is the dissipation coefficient. Spatial gradient operator, This indicates complex conjugation, and Im[] represents the imaginary part extraction operation.

[0033] A steady-state quantum turbulence generation system that preserves particle number conservation includes:

[0034] The parameter input module is configured to acquire the initial physical parameters of the quantum fluid system, which include at least the initial wave function, the interparticle interaction strength, and the chemical potential.

[0035] The model building module is configured to introduce an external driving term and a particle number conservation dissipation term into the Gross-Pitaevskii equation to build an evolutionary model of the system; wherein the external driving term is an additional external potential that varies in space and time, and the particle number conservation dissipation term depends on the time rate of change of particle number density, and both are configured to maintain the total number of particles in the system.

[0036] The evolution execution module is configured to perform numerical time evolution of the quantum fluid system based on the evolution model and initial physical parameters;

[0037] The steady-state determination and output module is configured to monitor the total kinetic energy of the quantum fluid system, determine that the system has entered a steady-state quantum turbulence state when the total kinetic energy reaches dynamic equilibrium, and output the distribution of quantum fluid physical quantities under the steady-state quantum turbulence state.

[0038] A computer device, the computer device comprising:

[0039] One or more processors;

[0040] Memory, used to store one or more programs;

[0041] When the one or more programs are executed by the one or more processors, the one or more processors implement the above-described steady-state quantum turbulence generation method that preserves particle number conservation.

[0042] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the above-described method for generating steady-state quantum turbulence while preserving particle number conservation.

[0043] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0044] This invention introduces a dissipation mechanism that depends on the time-varying rate of particle number density, enabling the quantum fluid system to still satisfy the continuity equation even in the presence of dissipation, without the need for chemical potential compensation. The external driving potential and the dissipation mechanism in this invention are physically independent of each other, and the energy injection scale, driving intensity and dissipation rate can be adjusted separately, thereby achieving flexible control over the quantum turbulence structure type and statistical characteristics. Attached Figure Description

[0045] Figure 1 This is a flowchart of the method in an embodiment of the present invention.

[0046] Figure 2 Total kinetic energy in the embodiments of the present invention The result of evolution over time.

[0047] Figure 3 The particle number density in steady-state quantum turbulence in this embodiment of the invention. The spacetime diagram.

[0048] Figure 4 The total number of particles in the embodiments of the present invention Changes over time. Detailed Implementation

[0049] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0050] A method for generating steady-state quantum turbulence while preserving particle number conservation includes the following steps:

[0051] An evolutionary model of a quantum fluid system is constructed, which is derived from the Gross-Pitaevskii equation by introducing an external driving term and a particle number conservation dissipation term; wherein:

[0052] The external driving term is an additional external potential that varies in space and time, used to inject energy into the system; the particle number conservation dissipation term depends on the time rate of change of particle number density, used to dissipate the energy of the system; both the external driving term and the particle number conservation dissipation term are configured to maintain the total particle number of the system.

[0053] The quantum fluid system is subjected to time evolution based on the aforementioned evolution model until the system reaches a steady-state quantum turbulence state with dynamic equilibrium of total kinetic energy.

[0054] The specific steps for constructing the evolutionary model of the quantum fluid system include:

[0055] (1) The evolution equation of quantum fluid is determined to be the Gross–Pitaevskii equation, specifically in the form of:

[0056]

[0057] in, The order parameter is the superfluid condensation wave function (hereinafter referred to as the wave function). Chemical potential For nonlinear interaction strength, Let be the evolution time. For a uniform static solution, the relationship is: .

[0058] The hydrodynamic form of the Gross-Pitaevskii equations can be obtained by performing a Madelung transform on the wave function, i.e. Received, among which , where is the particle number density. As the phase, its spatial gradient gives the superfluid velocity. Substituting this transformation into the Gross–Pitaevskii equations, separating the real and imaginary parts, yields the hydrodynamic form of the Gross–Pitaevskii equations, including the continuity equation and the Euler equations corrected for quantum pressure:

[0059]

[0060] in Corresponding to classical pressure, This refers to quantum pressure.

[0061] (2) Determine the driving method for the flow velocity to maintain particle number conservation during the driving process. Then, the fluid dynamics form under the applied driving force can be rewritten as:

[0062]

[0063] in This is the driving force that propels the flow velocity. The continuity equation remains unchanged, therefore the particle number is conserved.

[0064] Assuming the force is conservative, the driving force can be written as the gradient of the potential function, i.e. Substituting into Euler's equation, we get:

[0065]

[0066] Where the potential function and classical pressure The status is equivalent, therefore we revert to the Gross–Pitaevskii equations, where the driving force can appear as an external potential, i.e.

[0067]

[0068] By constructing a potential function V to obtain the driving force for spatiotemporal changes, the driving scale and intensity can be controlled. The specific steps are as follows:

[0069] Define the spatial scale and define the range of action of the driving force in the momentum space. Construct a piecewise function:

[0070]

[0071] in Where L is the wave number and L is the size of the spatial region. The minimum wavenumber of the driving force in momentum space. This represents the maximum wave number. It is a random phase, with a value range of 1. .

[0072] Further The fundamental potential field is obtained by inverse Fourier transform to coordinate space. Normalize its amplitude to ,in This represents the maximum driving force. Then, its time-varying behavior is set:

[0073]

[0074] Finally Renormalized to .in The correlation factor indicates that the driving force between two moments has a certain temporal correlation. The time period of the external potential can be used to control the rate of energy injection and the total energy at the final steady state, i.e., the driving intensity.

[0075] (3) Introduce a dissipation term dependent on the time-varying rate of particle number density to maintain particle number conservation. Write this into the Gross–Pitaevskii equation, specifically in the form:

[0076]

[0077] The second term on the left side of the equation is a dissipation term. is the dissipation coefficient, used to control the strength of dissipation. This form of dissipation ensures that the total energy decay is only related to the time rate of change of the particle number density, specifically:

[0078] ,

[0079] in Let be the total energy. The first term of the integrand is the linear energy, and the second term is the nonlinear interaction energy.

[0080] The dissipative Gross–Pitaevskii equations are transformed into a hydrodynamic form using the Madelung transformation.

[0081]

[0082] The continuity equation remains satisfied, meaning the particle number conservation is satisfied, and dissipation only acts in the Euler equation.

[0083] Furthermore, the dissipation term can be rewritten in spatial derivative form according to the continuity equation, specifically:

[0084] Using the Madelung transform, the continuity equation can be rewritten as ,in Indicates complex conjugation. This is an operation to extract the imaginary part.

[0085] Using this relationship, the dissipative Gross–Pitaevskii equation can be rewritten as follows:

[0086]

[0087] (4) Combining the external driving term with the above-mentioned particle number conservation dissipation term, the driving-dissipation evolution model is obtained, specifically as follows:

[0088]

[0089] By setting periodic boundary conditions for the evolution model, providing initial values, and setting chemical potential, interaction strength, and dissipation strength, the evolution model can be applied to the time evolution of a quantum fluid system until the system reaches a steady-state quantum turbulent state of dynamic equilibrium of total kinetic energy.

[0090] Example 1

[0091] To illustrate the application process and prediction results of the method in detail, the application of this invention will be explained in detail below using a specific practical situation as an example.

[0092] like Figure 1 As shown, for the quasi-one-dimensional case in space, the following steps are taken to generate steady-state quantum turbulence while preserving the particle number conservation:

[0093] Step 1: Determine the specific form of the Gross–Pitaevskii equation and the external potential driving method that preserves the particle number conservation, and set the parameters. .

[0094] Step 2: Set the space length Take the driving scale That is, the drive is applied to the two largest scales of the system. The drive cycle is taken. That is, a force is applied once every dimensionless time unit. The driving amplitude is taken as... .

[0095] Step 3: Add a dissipation term and set the dissipation coefficient. This corresponds to the case of weak dissipation.

[0096] Step 4: Define spatial discrete points for the complete driving dissipation equation. The Fourier spectrum method is used to set periodic boundary conditions.

[0097] Step 5: Take a uniform unit initial value. The fourth-order Jung-Kutta evolutionary scheme is adopted, with a time evolution step size of [value missing]. The long-term evolution results are as follows: Figure 2 , 3 As shown in Figure 4, kinetic energy Reaching steady state, particle number density The spacetime diagram reflects shock wave and soliton structure, total number of particles It remains unchanged.

[0098] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0099] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0100] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0101] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0102] Other embodiments of this application will readily occur to those skilled in the art upon consideration of the specification and practice of the disclosure herein. This application is intended to cover any variations, uses, or adaptations of this application that follow the general principles of this application and include common knowledge or customary techniques in the art not disclosed herein. The specification and embodiments are to be considered exemplary only, and the true scope and spirit of this application are indicated by the claims.

[0103] The above specific embodiments are used to explain and illustrate the present invention, but not to limit the present invention. Any modifications and changes made to the present invention within the spirit and scope of the claims shall fall within the protection scope of the present invention.

Claims

1. A method for generating steady-state quantum turbulence conserving the number of particles, characterized by, Includes the following steps: An evolutionary model of a quantum fluid system is constructed, which is derived from the Gross-Pitaevskii equation by introducing an external driving term and a particle number conservation dissipation term; wherein: The external driving term is an additional external potential that varies in space and time, used to inject energy into the system; the particle number conservation dissipation term depends on the time rate of change of particle number density, used to dissipate the energy of the system; both the external driving term and the particle number conservation dissipation term are configured to maintain the total particle number of the system. The quantum fluid system is subjected to time evolution based on the aforementioned evolution model until the system reaches a steady-state quantum turbulence state with dynamic equilibrium of total kinetic energy; The external driver is introduced in the following way: Based on the fluid dynamics form of the Gross-Pitaevskii equations, the flow velocity is driven by introducing a driving force into the Euler equations while keeping the continuity equations unchanged. The driving force is a conservative force, and the potential function corresponding to the driving force is introduced into the Gross-Pitaevskii equation as the additional external potential. By constructing a potential function to obtain the driving force of spatiotemporal changes, the driving scale and driving intensity can be controlled. The specific steps include: Setting the action scale range k of driving force in momentum space V , and constructing a piecewise function V k : ， wherein represents a generation method of driving force in momentum space, is a random phase in [0, 1], is a wave number, and L is a size of a spatial region; For V k Performing the inverse Fourier transform yields the fundamental potential field V in coordinate space. x and the fundamental potential field V x The amplitude is normalized to [0, V0], where This represents the maximum driving force. The potential function is updated using an iterative formula with time correlation: ， in, Let be the potential function at the current time t. is the potential function at the next time step after the update; T is the time variation period, used to control the rate of energy injection and the driving intensity; R is the correlation factor, used to control the degree of correlation between driving potentials at adjacent time steps; The amplitude of the potential function obtained after the update is renormalized to [0, V0].

2. The method of claim 1, wherein the method is a method of generating a steady-state quantum turbulent flow with preserved number of particles. The specific form of the particle number conservation dissipation term in the Gross–Pitaevskii equation is as follows: ， in, Let i be the superfluid condensation wave function, and i be an imaginary number. Let g be the chemical potential, g be the nonlinear interaction strength, and t be time. For spatial gradient operators; The particle number conservation dissipation term ensures that the total energy decay is only related to the time rate of change of particle number density, and λ is the dissipation coefficient used to control the dissipation intensity.

3. The method of claim 2, wherein the method is a method of generating a steady-state quantum turbulent flow with preserved particle number, characterized in that, Using the continuity equation, the particle number conservation dissipation term is constructed in spatial derivative form. The Gross–Pitaevskii equation with the particle number conservation dissipation term is expressed as: ， wherein denotes complex conjugation, and Im[] is the take imaginary part operation.

4. The method of claim 1, wherein the method is a method of generating a steady-state quantum turbulent flow with preserved number of particles. The evolutionary model is composed of the external driving term defined in claim 1 and the particle number conservation dissipation term defined in claim 2 or 3.

5. The method of claim 1, wherein the method is a method of generating a steady-state quantum turbulent flow with preserved number of particles, characterized in that, The evolutionary model is expressed as follows: ， in, Let i be the superfluid condensation wave function, and i be an imaginary number. Where is the chemical potential, g is the nonlinear interaction strength, and V(t,x) is the external driving term. For evolution time, In real space coordinates; Here, λ is the dissipation term for particle number conservation, and λ is the dissipation coefficient. Spatial gradient operator, This indicates complex conjugation, and Im[] represents the imaginary part extraction operation.

6. A particle number conserving steady-state quantum turbulent flow generating system, characterized by, include: The parameter input module is configured to acquire the initial physical parameters of the quantum fluid system, which include at least the initial wave function, the interparticle interaction strength, and the chemical potential. The model building module is configured to introduce an external driving term and a particle number conservation dissipation term into the Gross-Pitaevskii equation to build an evolutionary model of the system; wherein the external driving term is an additional external potential that varies in space and time, and the particle number conservation dissipation term depends on the time rate of change of particle number density, and both are configured to maintain the total number of particles in the system. The evolution execution module is configured to perform numerical time evolution of the quantum fluid system based on the evolution model and initial physical parameters; The steady-state determination and output module is configured to monitor the total kinetic energy of the quantum fluid system, determine that the system has entered a steady-state quantum turbulence state when the total kinetic energy reaches dynamic equilibrium, and output the distribution of quantum fluid physical quantities under the steady-state quantum turbulence state. The external driver is introduced in the following way: Based on the fluid dynamics form of the Gross-Pitaevskii equations, the flow velocity is driven by introducing a driving force into the Euler equations while keeping the continuity equations unchanged. The driving force is a conservative force, and the potential function corresponding to the driving force is introduced into the Gross-Pitaevskii equation as the additional external potential. By constructing a potential function to obtain the driving force of spatiotemporal changes, the driving scale and driving intensity can be controlled. The specific steps include: Setting the range of the action scale k of the driving force in momentum space V and constructing a piecewise function V k : ， in This indicates how the driving force is generated in momentum space. A random phase within [0, 1] Where L is the wave number and L is the size of the spatial region; V k The inverse Fourier transform is performed to obtain the basis potential field V x in the coordinate space, and the amplitude of the basis potential field V x is normalized to [0, V0], wherein is the maximum value of the driving force. The potential function is updated using an iterative formula with time correlation: ， wherein, is the potential function at the current time t, is the potential function at the next time after updating; T is the time variation period, used to control the speed of energy injection and the driving strength; R is the correlation factor, used to control the correlation degree between the driving potentials of adjacent times. The amplitude of the potential function obtained after the update is renormalized to [0, V0].

7. A computer device, characterized by The computer device includes: One or more processors; Memory, used to store one or more programs; When the one or more programs are executed by the one or more processors, the one or more processors implement the steady-state quantum turbulence generation method that preserves particle number as described in any one of claims 1-5.

8. A computer readable storage medium having stored thereon a computer program, characterized in that, When the program is executed by the processor, it implements a steady-state quantum turbulence generation method that preserves particle number as described in any one of claims 1-5.

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