The thesis presents a systematic study of the Mpemba effect in a colloidal system with a micron-sized particle diffusing in a water bath. While the Mpemba effect, where a system’s thermal relaxation time is a non-monotonic function of the initial temperature, has been observed in water since Aristotle’s era, the underlying mechanism of the effect is still unknown. Recent studies indicate that the effect is not limited to water and has been studied both experimentally and numerically in a wide variety of systems. By carefully designing a double-well potential using feedback-based optical tweezers, the author demonstrates that an initially hot system can sometimes cool faster than an initially warm system. The author also presents the first observation in any system of another counterintuitive effect―the inverse Mpemba effect―where the colder of the two samples reaches the thermal equilibrium at a hot temperature first. The results for both the observations agree with theoretical predictions based on the Fokker-Planck equation. The experiments reveal that, for carefully chosen conditions, a strong version of both of the effects are observed where a system can relax to the bath temperature exponentially faster than under typical conditions.
Author(s): Avinash Kumar
Series: Springer Theses
Publisher: Springer
Year: 2022
Language: English
Pages: 134
City: Cham
Supervisor's Foreword
Acknowledgments
Contents
Parts of This Thesis Have Been Published in the Following Journal Articles
1 Introduction
1.1 History of the Mpemba Effect
1.2 Explanations for the Mpemba Effect
1.3 Mpemba Effect in Other Systems
1.3.1 Experiments
1.3.2 Numerical Studies
1.4 Mpemba Effect in Colloidal Systems
1.5 Particle Manipulation Techniques
1.5.1 Passive Trapping
Optical Tweezers
Magnetic Tweezers
Holographic Tweezers
1.5.2 Active Trapping
Electrokinetic Traps
Hydrodynamic Traps
Acoustic Traps
Thermal Traps
1.6 Combining Feedback Traps and Optical Tweezers
1.7 Overview of the Thesis
References
2 Particle Dynamics
2.1 The Langevin Equation
2.1.1 A Free Particle
2.1.2 A Trapped Particle
2.2 Fokker–Planck Equation
2.2.1 Adjoint of the Fokker–Planck Operator
2.2.2 Eigenfunctions and Eigenvalues of the Fokker–Planck Operator
2.2.3 Fokker–Planck Equation with no Drift
2.3 Heat Equation
2.4 Supplementary Information
2.4.1 A Similarity Transformation of the Fokker–Planck Operator
References
3 Optical Feedback Traps
3.1 Principles of Optical Tweezers
3.2 Optical Tweezers Setup
3.2.1 Faraday Isolator
3.2.2 Acousto-Optic Deflector
3.2.3 Detection Scheme
3.2.4 Control and Data Acquisition
3.3 Sample Preparation
3.4 Calibration
3.4.1 Position Calibration
3.4.2 Trap-Stiffness Calibration
3.5 Virtual Harmonic Potential
3.6 Isotropic Traps
3.7 Virtual Double-Well Potential
3.8 Discussion
References
4 Mpemba Effect
4.1 Definition of the Mpemba Effect
4.2 Energy Landscape for the Mpemba Effect
4.2.1 Choice of Potential Energy Landscape
4.3 Imposing an Instantaneous Quench via Initial Conditions
4.4 Measuring the Distance to Equilibrium
4.4.1 L1 distance Distance
4.4.2 Kullback–Leibler (KL) Divergence
4.5 Observation of the Mpemba Effect in Asymmetric Domains
4.6 Analysis Based on Eigenfunction Expansion
4.6.1 Calculation of the a2 Coefficient
4.6.2 Relationship Between D and the a2 Coefficient
4.7 Strong Mpemba Effect
4.8 Geometric Interpretation of the Mpemba Effect
4.8.1 Thermalization in a Double-Well Potential with Metastability
4.8.2 Metastable Mpemba Effect
4.8.3 Metastable Mpemba Effect in Terms of Extractable Work
4.9 Discussion
4.10 Supplementary Information
4.10.1 Infinite Potential vs. Finite Potential
4.10.2 Calculation of Equilibration Time
4.10.3 Equilibration Time Versus the a2 Coefficient
4.10.4 Barrier Height vs. Discontinuity in Local Equilibrium
References
5 Inverse Mpemba Effect
5.1 Energy Landscape for the Inverse Mpemba Effect
5.2 Inverse Mpemba Effect in an Asymmetric Potential
5.3 Analysis Based on Eigenfunction Expansion
5.4 Discussion
References
6 Higher-Order Mpemba Effect
6.1 Experiment
6.2 Eigenfunction Analysis
6.3 Mpemba Effect in a Potential with One Local Minimum
6.4 Discussion
Reference
7 Conclusions
7.1 Summary of the Results Obtained
7.2 Final Remarks
References
