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## Appendix A 

## Derivation of Thermal Fluctuation of φ 

Our aim is to calculate the thermal fluctuation of φ that exists when the field is in thermal equilibrium with a thermal bath. This derivation has been worked through following Ref. [56]. As we are dealing with the thermal fluctuation of φ about φ = 0 , we will simply call δφ by φ . 

The solution to the perturbation equation 

is 

where 

<img src='content_image/323982.jpg'>

<img src='content_image/323981.jpg'>

<img src='content_image/323983.jpg'>

