Finding Discrete Counterpart of Continuous Variable Energy Term

[saravanan13]

Dear Sir…
I am looking for a discrete counter part of a continuous variable.
the continuous version of energy term in a liquid crystal is given by [\vec{n}\cdot(\nabla\times\vec{n})]^2. This is a square of a dot product between a vector 'n' and its curl field. My question is what is the exact discrete version of this energy term?
In a literature it is given as [\hat{k}\cdot(\vec{n}_{i}\times\vec{n}_{i+1})]^2, it is mentioned that \hat{k} is a unit vector, \vec{n}_{i} is a vector at lattice point 'i' and \vec{n}_{i+1} is a vector at the lattice point 'i+1'.

For your clarification I have attached a picture file. See second term in Eq. (1) and below theory for justification in the attached file.

Thanks in advance.

 

[jvicens]

Thank you for your inquiry regarding the discrete counterpart of a continuous variable in the energy term for liquid crystals. As a scientist in this field, I am happy to provide you with an explanation and further clarification.

The discrete version of the energy term you mentioned is indeed given by [\hat{k}\cdot(\vec{n}_{i}\times\vec{n}_{i+1})]^2, where \hat{k} is a unit vector, \vec{n}_{i} is a vector at lattice point 'i' and \vec{n}_{i+1} is a vector at the lattice point 'i+1'. This can be derived from the continuous version by discretizing the space into a lattice and approximating the continuous derivatives with finite differences.

To understand this further, let's look at the theory behind it. In liquid crystals, the molecules are arranged in a specific direction called the director, represented by the vector \vec{n}. The energy of the system is affected by the orientation of this director, which can be described by its curl field, \nabla\times\vec{n}. In the continuous version, the energy is given by [\vec{n}\cdot(\nabla\times\vec{n})]^2, which represents the square of the dot product between \vec{n} and its curl field.

Now, in the discrete version, we can approximate the curl field at a lattice point by taking the difference between the director at that point and the director at the neighboring point, i.e. \nabla\times\vec{n} \approx \vec{n}_{i+1} - \vec{n}_{i}. This leads to the discrete version of the energy term, [\hat{k}\cdot(\vec{n}_{i}\times\vec{n}_{i+1})]^2, where \hat{k} is a unit vector perpendicular to the plane formed by \vec{n}_{i} and \vec{n}_{i+1}. This term represents the alignment of the directors at neighboring lattice points and is a good approximation of the continuous energy term.

I hope this explanation helps you understand the logic behind the discrete version of the energy term. If you have any further questions, please do not hesitate to ask. Thank you for your interest in this topic.



Scientist in Liquid Crystal Research