Since the original proposal by Thouless , the geometrical pumping has been studied extensively. It is remarkable that the observed pumping current in experiments exists even if there is no average bias [2,3]. We have recognized that the essence of the geometrical pumping is Berry’ phase under adiabatic operations. Contrast to the original idea by Berry, the Berry- Sinitsyn-Nemenman (BSN) curvature can be used even in classical systems if we have multiple control parameters . If the BSN curvature exists in strongly non-equilibrium situations, Sagawa and Hayakawa shows that the entropy can be a path-dependent quantity [6,7]. We also develop the analysis of quantum master equation for geometrical pumping and obtain the pumping current in adiabatic pumping process . In this talk, in addition to our previous studies on the average current in the adiabatic geometrical pumping, we discuss how the fluctuation theorem has been changed in this system, in which non-Gaussian geometric fluctuations play important roles. We also discuss how the result is changed if we include the non-adiabatic effects [9,10].
 D. J. Thouless, Phys. Rev. B 27, 6083 (1983).
 H. Pothier, P. Lafarge, C. Urbina, D. Esteve, and M. H. Devoret, EPL. 17, 249 (1992).
 M. Switkes, C. M. Marcus, K. Campman, and A. C. Gossard, Science 283, 1905 (1999).
 M. V. Berry, Proc. R. Soc. London A 392, 45 (1984).
 N. A. Sinitsyn and I. Nemenman, EPL 77, 58001 (2007), Phys. Rev. Lett. 99, 220408 (2007).
 T. Sagawa and H. Hayakawa, Phys. Rev. E 84, 051110 (2011).
 T. Yuge, T. Sagawa, A. Sugiura, and H. Hayakawa, J. Stat. Phys. 153, 412 (2013).
 T. Yuge, T. Sagawa, A. Sugita, and H. Hayakawa, Phys. Rev. B 86, 235308 (2012).
 K. L.Watanabe and H. Hayakawa, Prog. Theor. Exp. Phys. 2014, 113A01 (2014).
 K. L.Watanabe and H. Hayakawa, Phys. Rev. E 96, 022118 (2017).