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With the Schwinger's proper-time formalism of the Nambu-Jona-Lasinio model,we investigate the finite volume effects in the presence of magnetic fields.Since the coupling constant $G$ can be influenced by strong magnetic fields,the model is solved with a running coupling constant $G(B)$ which is fitted bythe lattice average $(\Sigma_u+\Sigma_d)/2$ and difference $\Sigma_u-\Sigma_d$.The investigation mainly focuses on the constituent quark mass and the thermalsusceptibility depending on the magnetic fields, the temperatures and thefinite sizes. For the model in finite or infinite volume, the magnetic fieldscan increase the constituent quark mass while the temperatures can decrease itinversely. There is a narrow range of the box length that makes the effects offinite volume perform prominently. The model will behave close to infinitevolume limit for larger box length. It is shown that the influence of finitevolume can be changed by magnetic fields and temperatures. Finally, we discussthe thermal susceptibility depending on the temperature in finite volume in thepresence of magnetic fields.

Finite volume effects of the Nambu-Jona-Lasinio model with the running coupling constant