Kinetic Entropy Increase and Macroscopic Dissipation: On the Energy Mechanism of Continuous Fluid Transport

Author: Zhang Suhang
Luoyang School of Mathematics

Abstract

Macroscopic fluid flows possess an intrinsic property of irreversible dissipation, and all continuous transport processes are accompanied by energy loss. Classical fluid mechanics introduces viscous dissipation only phenomenologically without clarifying its microscopic origin. In the Boltzmann–Grad limit, Deng Yu, Hani and Ma Xiao rigorously established the long-time derivation from hard-sphere particle dynamics to the Boltzmann equation, and further derived the Navier–Stokes–Fourier equation via the fluid-dynamic limit (Chapman–Enskog expansion). This chain reveals a structural picture: after statistical coarse-graining of reversible microscopic collisions, entropy growth and the time arrow described by Boltzmann’s H-theorem emerge at the mesoscopic level, while viscous dissipation appears at the macroscopic end of this chain as a product of statistical coarse-graining. Drawing on this picture, this paper establishes a cross-scale correspondence between microscopic statistical entropy growth and macroscopic fluid dissipation: entropy growth emerges from the statistical coarse-graining of reversible microscopic dynamics, and this process corresponds at the macroscopic scale to viscous dissipation and turbulent energy decay in the Navier–Stokes system. Combined with the fluid energy decomposition framework developed herein, a core engineering phenomenon can be explained theoretically: steady-state fluid transport requires continuous external pressurization and energy input to offset the inherent entropy loss of the flow. This paper achieves self-consistent connection among microscopic kinetic mechanisms, macroscopic fluid equations and engineering fluid laws.

Keywords: Boltzmann equation; kinetic limit; entropy increase; viscous dissipation; fluid energy decomposition; fluid transport

I. Introduction

Fluid flow is not a self-sustaining conservative motion; its natural evolution is always accompanied by energy loss. In engineering practice, pipeline water delivery, gas transport and circulating flows all require continuous work and pressurization by pumps or fans. The underlying theoretical origin of this engineering phenomenon has long lacked a unified cross-scale interpretation.

At the macroscopic level, the Navier–Stokes equation characterizes energy dissipation via the viscous term, yet it cannot explain why dissipation necessarily exists and cannot be eliminated. At the microscopic kinetic level, Boltzmann’s H-theorem establishes the irreversibility of entropy growth for statistical systems. Results obtained by Deng Yu, Hani and Ma Xiao rigorously derive the Boltzmann equation from hard-sphere particle dynamics in the Boltzmann–Grad limit over long time intervals, and further obtain the Navier–Stokes–Fourier equation from the Boltzmann equation under the fluid-dynamic limit.

Accordingly, this paper connects the structural picture of “reversible microdynamics → irreversible macroscopic entropy growth” revealed by the above kinetic-limit results with macroscopic energy decomposition rules and engineering transport phenomena, to construct a self-consistent theoretical chain of fluid dissipation.

II. Microscopic Kinetic System: From Reversible Microphysics to Irreversible Macrophysics

Newtonian collisions of pure microscopic particles are time-reversible, energy-conserving and free of inherent loss. For statistical systems composed of a large number of particles under the Boltzmann–Grad scaling limit (particle number tends to infinity, particle diameter tends to zero, with their product kept finite), the evolution follows the Boltzmann equation. Within this scaling framework, the work of Deng Yu, Hani and Ma Xiao extends Lanford’s short-time result to the full time interval where strong solutions of the Boltzmann equation exist, completing the rigorous derivation from hard-sphere dynamics to the Boltzmann equation.

Boltzmann’s H-theorem proves that the entropy of a closed statistical system is non-decreasing, the evolution carries a unique time arrow, and the system spontaneously evolves toward energy dispersion. This irreversible statistical process no longer obeys the time-reversal symmetry of microscopic dynamics. Entropy growth and the time arrow emerge at this micro-to-mesoscopic level.

On this basis, through Chapman–Enskog multi-scale expansion, the macroscopic moment equations of the Boltzmann equation converge to the compressible Euler equations and the incompressible Navier–Stokes–Fourier equations. The viscosity coefficient emerges inevitably from this expansion at the meso-to-macro level, rather than being an artificially added phenomenological parameter. This leads to the core conclusion:

The viscous dissipation term in the Navier–Stokes equation has its microscopic counterpart in entropy growth emerging from statistical coarse-graining; this picture is rigorously supported within the dilute-gas chain.

A boundary note is necessary: the rigorous results for the first layer (Boltzmann–Grad limit) are restricted to dilute gases and cannot be directly extended without qualification to dense fluids and liquids. This paper adopts only the structural picture disclosed by this chain — that irreversible macroscopic entropy growth emerges from statistical coarse-graining of reversible microscopic dynamics — instead of transplanting the rigorous dilute-gas conclusions directly to liquids. Rigorous kinetic descriptions of liquids generally rely on the Enskog equation or denser theories; nevertheless, the structural relation that “macroscopic dissipation corresponds to irreversible statistical processes at the microscopic level remains valid for liquids.

III. Macroscopic Energy Decomposition for Real Fluids

Based on the dual-axiom fluid system established in previous work, the energy distribution of real fluids satisfies the unified relation:

\Delta p \sim \Delta\left(\frac12\rho v_\parallel^2\right)+E_{\text{turb}}+\Phi_{\text{diss}}

Notation:

- \Delta\left(\frac12\rho v_\parallel^2\right): change in tangential kinetic energy sustaining the mainstream flow;
- E_{\text{turb}}: stored kinetic energy of turbulent fluctuations in flow perturbations;
- \Phi_{\text{diss}}: irreversible viscous dissipation energy;
- \sim denotes an equivalent energy partition relation for the system.

Combined with the kinetic entropy-growth mechanism above, it follows that:

1. \Phi_{\text{diss}} directly corresponds to immediate dissipation associated with Boltzmann entropy growth at the microscopic scale;
2. E_{\text{turb}} is an intermediate mode storing energy, which eventually breaks down via vortex cascades into small-scale viscous dissipation, representing delayed entropy loss.

Therefore, apart from sustaining directional motion, most of the energy supplied by the pressure difference continuously dissipates through either immediate or delayed entropy-growth processes. Macroscopically, viscous dissipation and turbulent dissipation are two manifestations of the same entropy-growth process: the former is immediate, the latter delayed, and both ultimately correspond to the irreversible conversion of mechanical energy into internal energy.

IV. Theoretical Correspondence: Entropy Growth as the Underlying Cause of Energy Consumption in Fluid Transport

The ideal inviscid Bernoulli system conserves energy perfectly, with no loss or decay, so external energy input is unnecessary to sustain flow. However, this model is an extreme idealized case that excludes entropy growth and does not describe real fluids.

For real physical systems:
Real fluid = directional kinetic-energy system + continuous entropy-dissipation system

The irreversibility of entropy growth dictates that any sustained fluid flow inevitably consumes mechanical energy, which disperses as internal energy. A flow cannot maintain itself indefinitely using its own energy; velocity attenuation, weakening of flow structures and eventual stagnation will occur. The kinetic derivations by Deng Yu and coauthors provide a rigorous dilute-gas example for the structural picture of “reversible microdynamics → irreversible macroscopic entropy growth”, thereby offering cross-scale support for the origin of this irreversible dissipation.

V. Engineering Mechanism: The Physical Nature of Pump Pressurization

From the above cross-scale theory, the core rule of fluid engineering can be interpreted:

Continuous pressurization and work input by pumps and fans essentially supply pressure potential persistently to offset the inherent entropy dissipation of the system.

The balance condition for steady fluid transport reads:

Rate of external energy input = rate of immediate viscous dissipation + rate of delayed turbulent dissipation

Without continuous energy input from a pump, entropy dissipation will deplete the kinetic energy of the flow field until all macroscopic directional flow vanishes.

This theoretically demonstrates that self-sustained steady transport with zero energy consumption does not exist for real fluids. Continuous pressurization is a fundamental means to counteract entropy growth and maintain flow. This conclusion follows directly from the first and second laws of thermodynamics and does not depend on details of any specific kinetic model.

VI. Summary

1. Microscopic particle dynamics is reversible and energy-conserving, while macroscopic fluid dissipation is irreversible. In the Boltzmann–Grad limit, Deng Yu, Hani and Ma Xiao rigorously derived the Boltzmann equation from hard-sphere dynamics over long time intervals; entropy growth and the time arrow emerge at this stage. Subsequent derivation of the N-S-F equations via the fluid-dynamic limit shows that viscous dissipation arises at the macroscopic end of this chain as a product of statistical coarse-graining.
2. Viscous dissipation and turbulent energy decay in the Navier–Stokes equations are essentially macroscopic manifestations of entropy growth at different levels: the former immediate, the latter delayed, both converging ultimately to the irreversible conversion of mechanical energy into internal energy.
3. The energy decomposition formula for real fluids fully characterizes the pathways of energy distribution and dissipation, unifying the theoretical boundary between ideal conservative flows and real dissipative flows.
4. The underlying mechanism for continuous pump pressurization in engineering is to offset the inherent entropy loss of the fluid by external energy input and sustain macroscopic directional steady transport. This conclusion follows from thermodynamic laws and is independent of specific kinetic models.

This paper establishes a four-level connection: kinetic-limit results → microscopic entropy-growth picture → macroscopic N-S dissipation structure → engineering fluid transport laws. It supplements a cross-scale underlying logical thread for classical fluid mechanics.