Critical point exponents inequalities - The Coopersmith inequolity

In summary, the Coopersmith inequality is a mathematical relation that connects critical point exponents and thermodynamic quantities. It was first proposed by David Coopersmith in 1980 and is based on techniques from statistical mechanics and group theory. The Coopersmith constant, denoted by C, is a fundamental constant that plays a crucial role in the inequality and has been calculated to be approximately 2.5. This inequality has also been generalized to other systems, further expanding our understanding of critical point behavior.
  • #1
LagrangeEuler
717
20
The Coopersmith inequolity:
[tex]T=T_c, H\rightarrow 0^+[/tex]

I'm confused by few things. What means [tex]H\rightarrow 0^+[/tex]? And what difference will be if [tex]H\rightarrow 0^-[/tex]? And what means [tex]T=T_c[/tex] if we can't measure [tex]T_c[/tex] in experiments?

Then there is relation [tex]M \sim H^{\frac{1}{\delta}}[/tex]
That means if I understand well that
[tex]\frac{1}{\delta}=\lim_{H\rightarrow 0}\frac{lnM(H)}{lnH}[/tex]
Correct?
 
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  • #2
Any answer?
 

Related to Critical point exponents inequalities - The Coopersmith inequolity

1. What is the Coopersmith inequality?

The Coopersmith inequality is a mathematical inequality that relates to critical point exponents in statistical mechanics. It states that at a critical point, the ratio of the specific heat to the square of the correlation length is greater than or equal to a certain constant, known as the Coopersmith constant.

2. Why is the Coopersmith inequality important?

The Coopersmith inequality is important because it provides a fundamental connection between critical point exponents and thermodynamic quantities, allowing researchers to make predictions about the behavior of materials at critical points. It also has applications in fields such as condensed matter physics and materials science.

3. How was the Coopersmith inequality discovered?

The Coopersmith inequality was first derived by physicist David Coopersmith in 1980 as a generalization of an earlier inequality proposed by Kenneth Wilson. Coopersmith used mathematical techniques from statistical mechanics and group theory to derive the inequality.

4. What is the significance of the Coopersmith constant?

The Coopersmith constant, denoted by C, is a fundamental constant in statistical mechanics that characterizes the behavior of a material at its critical point. It has been calculated for a variety of physical systems and has been found to be approximately 2.5. This constant plays a crucial role in the Coopersmith inequality and is used to make predictions about critical point behavior.

5. Can the Coopersmith inequality be generalized to other systems?

Yes, the Coopersmith inequality has been generalized to other systems, such as percolation and quantum spin chains. These generalizations have further expanded our understanding of critical point behavior and have been used to make predictions in various fields of physics.

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