Appendix C.3. Random electric field corrections References 12 In the presence of an external electric field (53), the transition probabilities (56) become The Kronecker deltas now also allow next-neighbor transitions m = n GLYPH<6> 1. Averaging over the noise GLYPH<21>$_{t}$ using Eq. (54), we finally obtain, $^{D}$p GLYPH<28> m ; n$^{E}$ GLYPH<21>$_{t}$ =$^{GLYPH<12>}$ GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<14>$_{m}$$_{;}$$_{n}$ + 1 i$_{ℏ}$ GLYPH<28> Z 0 dt exp $^{GLYPH<0>}$i !$_{m}$$_{;}$$_{n}$ t GLYPH<1> GLYPH<2>$^{(}$ GLYPH<0> ℏ 4 !$_{0}$ $^{GLYPH<16>}$! 2 0 GLYPH<0> ! 2 t GLYPH<17> GLYPH<20> p n + 1 p n + 2 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{+}$$_{2}$ + (2 n + 1) GLYPH<14>$_{m}$$_{;}$$_{n}$ + p n p n GLYPH<0> 1 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{GLYPH<0>}$$_{2}$ GLYPH<21> ) GLYPH<12> GLYPH<12> GLYPH<12> 2 + 1 2 M $_{ℏ}$!$_{0}$ $^{GLYPH<2>}$( n + 1) GLYPH<14>$_{m}$$_{;}$$_{n}$$_{+}$$_{1}$ + n GLYPH<14>$_{m}$$_{;}$$_{n}$$_{GLYPH<0>}$$_{1}$ GLYPH<3> GLYPH<2> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<28> Z 0 dt GLYPH<28> Z 0 ds exp $^{GLYPH<0>}$i !$_{m}$$_{;}$$_{n}$ ( t GLYPH<0> s ) GLYPH<1> h GLYPH<21>$_{t}$ GLYPH<21>$_{s}$ i$^{GLYPH<12>}$ GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> : (C.5) p GLYPH<28> m ; n =$^{GLYPH<12>}$ GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<14>$_{m}$$_{;}$$_{n}$ + 1 i$_{ℏ}$ GLYPH<28> Z 0 dt exp $^{GLYPH<0>}$i !$_{m}$$_{;}$$_{n}$ t GLYPH<1> GLYPH<2>$^{(}$ GLYPH<0> ℏ 4 !$_{0}$ $^{GLYPH<16>}$! 2 0 GLYPH<0> ! 2 t GLYPH<17> GLYPH<20> p n + 1 p n + 2 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{+}$$_{2}$ + (2 n + 1) GLYPH<14>$_{m}$$_{;}$$_{n}$ + p n p n GLYPH<0> 1 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{GLYPH<0>}$$_{2}$ GLYPH<21> + GLYPH<21>$_{t}$ s ℏ 2 M !$_{0}$ GLYPH<18> p n + 1 GLYPH<14>$_{m}$$_{;}$$_{n}$$_{+}$$_{1}$ + p n GLYPH<14>$_{m}$$_{;}$$_{n}$$_{GLYPH<0>}$$_{1}$ GLYPH<19> ) GLYPH<12> GLYPH<12> GLYPH<12> 2 : (C.4) [1] A. Fennimore, T. Yuzvinsky, W. Han, M. Fuhrer, J. Cumings and A. Zettl, Nature 424 , 408 (2003). [12] S. Mukamel, Phys. Rev. Lett. 90 , 170604 (2003). [11] H. Tasaki, e-print arXiv:cond-mat / 0009244v2. [10] V. Blickle, T. Speck, L. Helden, U. Seifert, and C. Bechinger, Phys. Rev. Lett. 96 , 070603 (2006). [9] F. Douarche, S. Ciliberto, A. Petrosyan, and I. Rabbiosi, Euro- phys. Lett. 70 , 593 (2005). [8] J. Liphardt, S. Dumont, S. Smith, I. Tinoco Jr. and C. Busta- mante, Science 296 , 1832 (2002). [7] C. Jarzynski, Eur. Phys. J. B 64 , 331 (2008). [6] C. Jarzynski, Phys. Rev. Lett. 78 , 2690 (1997). [5] G. Gallavotti and E.G.D. Cohen, Phys. Rev. Lett. 74 , 2694 (1995). [4] D.J. Evans, E.G.D. Cohen, and G.P. Morriss, Phys. Rev. Lett. 71 , 2401 (1993). [3] P. H¨anggi and F. Marchesoni, Rev. Mod. Phys. 81 , 387 (2009). [2] G. Cerefolini, Nanoscale Devices , (Springer, Berlin, 2009). [32] M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions (Washington: United States department of commerce, 1972). [13] P. Talkner, M. Campisi, and P. H¨anggi, J. Stat. Mech. P02025 (2009). [14] M. Campisi, P. Talkner, and P. H¨anggi, Phys. Rev. Lett. 102 , 210401 (2009). [15] P. Talkner, E. Lutz and P. H¨anggi, Phys. Rev. E 75 , 50102(R) (2007). [16] G. Huber, F. Schmidt-Kaler, S. De GLYPH<11> ner and E. Lutz, Phys. Rev. Lett. 101 , 70403 (2008). [17] S. De GLYPH<11> ner and E. Lutz, Phys. Rev. E 77 , 021128 (2008). [18] D. Leibfried, R. Blatt, C. Monroe, D. Wineland, Rev. Mod. Phys. 75 , 281 (2003). [19] K. Husimi, Prog. Theo. Phys. 9 , 381 (1953). [20] K. Takayama, Phys. Rev. A 45 , 2618 (1992). [21] R. Kulsrud, Phys. Rev. 106 , 205 (1957). [22] C. Jarzynski, Phys. Rev. E 56 , 5018 (1997). [23] R. van Zon, L. Hern´andez de la Pe˜na, G. Peslherbe and J. Schofield, Phys. Rev. E 78 , 41103 (2008); ibid 78 , 41104 (2008). [24] F. Galve and E. Lutz, Phys. Rev. A 79 , 055804 (2009). [25] A. Messiah, Quantum Mechanics , (North-Holland, Amsterdam, 1961). [26] G. Huber, T. Deuschle, W. Schnitzler, R. Reichle, K. Singer and F. Schmidt-Kaler, New J. Phys. 10 , 013004 (2008). [27] C. Cohen-Tannoudji, B. Diu, F. Lalo¨e, Quantum Mechanics, Vol. 1,2 (Paris: Hermann, 1997). [28] Q. Turchette, D. Kielpinski, B. King, D. Leibfried, D. Meekhof, C. Myatt, M. Rowe, C. Sackett, C. Wood, W. Itano, C. Monroe and D. Wineland, Phys. Rev. A 61 , 063418 (2000). [29] T.A. Savard, K.M. O’Hara, and J.E. Thomas, Phys. Rev. A 56 , R1095 (1997). [30] K. Takayama, J. Math. Phys. 27 , 1747 (1986). [31] K. Takayama, Phys. Rev. A 34 , 4408 (1986).