Relaxation Data Analysis Using Heavy-Tail Function Fitting
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Current methods for analyzing relaxation responses in physical phenomena often misinterpret underlying characteristics due to inadequate mathematical descriptions, leading to incorrect interpretations and hindering technological progress.
Innovation Solution
A system and method that convert relaxation response data into linear-amplitude versus log-time data, perform a least-squares fit to a heavy-tail function to determine fit parameter values, and generate confidence intervals, allowing for accurate characterization of relaxation processes with fewer parameters.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If traditional mathematical descriptions are used to analyze relaxation responses, then the analysis method is simple, but the interpretation is incorrect and lacks accuracy
Solution Approach 1:
The patent transforms the relaxation response analysis by changing the mathematical parameters used - specifically using a heavy-tailed function with power-law decay instead of traditional exponential decay models. This parameter change allows accurate characterization of relaxation processes while maintaining manageable model complexity through physically motivated assumptions about the underlying mechanisms.
Solution Approach 2:
The patent introduces an intermediary mathematical framework that connects experimental relaxation data to physical mechanisms. The heavy-tailed function serves as an intermediary that bridges the gap between simple exponential models and complex physical reality, enabling accurate interpretation without requiring overly complicated models.
2Measurement precision
If more parameters are used to fit relaxation data, then the fit accuracy improves, but the model complexity increases
Solution Approach 1:
The patent applies partial action by using a simplified heavy-tailed function that captures the essential features of relaxation processes without requiring complete description of all underlying mechanisms. This approach achieves sufficient fit accuracy for practical applications while avoiding the complexity of fully comprehensive models.
Solution Approach 2:
By changing from traditional multi-parameter exponential models to a heavy-tailed function with power-law decay, the patent reduces the number of parameters needed while maintaining or improving fit accuracy. The power-law form naturally captures the broad time-scale behavior of relaxation processes with fewer parameters.
3Loss of information
If traditional relaxation analysis methods are used, then the analysis process is quick, but the physical insights are lost
Solution Approach 1:
The heavy-tailed function serves as an intermediary that preserves physical insights while maintaining analytical tractability. The function form directly relates to physical mechanisms (power-law decay, heavy tails) allowing interpretation of relaxation processes in terms of underlying physics without requiring complex computational analysis.
Solution Approach 2:
By using parameters with direct physical meaning (power-law exponent, characteristic time scales) instead of abstract fitting parameters, the method maintains quick analysis while preserving physical insights. The parameter changes enable direct connection between mathematical fit and physical mechanisms.
Data Source
AI summary
A system to improve a product based on a relaxation response includes a memory configured to store relaxation response data of a sample. The relaxation response data includes time data and amplitude data. A processor is operatively coupled to the memory and configured to convert the relaxation response data to linear-amplitude versus log-time data. The processor also performs a least-squares fit of the converted relaxation response data to a heavy-tail function to determine one or more fit parameter values. The processor also updates a design for the sample based at least in part on the one or more fit parameter values.