Agradecimentos: The authors would like to thank T. Lappi for pointing out the similarity of the reduction of dissipative transport coefficients in a magnetic field observed here to the mechanism suggested in Ref. [40]. G. S. D. greatly acknowledges the warm hospitality of the Department of Physics of Goethe University, where part of this work was done. E. M. and D. H. R. greatly acknowledge the warm hospitality of the Department of Physics of the University of Jyvaskyla, where part of this work was done. This work was supported by the Collaborative Research Center CRC-TR 211 "Strong-interaction matter under extreme conditions" funded by DFG. G. S. D. and J. N. thank Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq) for financial support. X. G. H. is supported by the Young 1000 Talents Program of China, NSFC with Grants No. 11535012 and No. 11675041. E. M. is supported by the Bundesministerium fur Bildung und Forschung (BMBF) and by the Research Council of Norway, (NFR) Project No. 255253/F50. H. N. is supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 655285 and by the Academy of Finland, Project No. 297058. J. N. and G. M. M. thank Fundacao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP) under Grants No. 2015/50266-2 (2017/05685-2) and No. 2016/13517-0, respectively, for financial support. D. H. R. is partially supported by the High-end Foreign Experts Project No. GDW20167100136 of the State Administration of Foreign Experts Affairs of China

Agradecimentos: We are grateful to Sriram Ganeshan and Paul Wiegmann for many fruitful discussions and suggestions related to this project. A. G. A.'s work was supported by Grant No. NSF DMR-1606591 and US DOE DESC-0017662. G. M. M. thanks Fundacao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP) for financial support under Grant No. 2016/13517-0

Agradecimentos: We are grateful to Paul Wiegmann for many discussions and for careful reading and commenting on an earlier draft of this work. A.G.A.'s research was supported by NSF Grant No. DMR-1606591 and U.S. DOE Grant No. DESC-0017662. A.G.A. and G.M.M. are grateful to the International Institute of Physics, Natal, Brazil, for hospitality. G.M.M. thanks Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) for financial support under Grant No. 2016/13517-0. S.G. acknowledges support from a PSC-CUNY Award. S.G. acknowledges the Aspen Center for Physics where part of this work was carried out, which is supported by National Science Foundation Grant No. PHY-1607611

Agradecimentos: We acknowledge Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for financial support. T.V.T. and G.M.M. were supported by FAPESP under Projects No. 2015/21349-7 and No. 2016/13517-0, respectively, whereas A.O.C. was supported by CNPq under Project No. 302420/2015-0

Abstract: We present an exact solution for the heat conductance along a harmonic chain connecting two reservoirs at different temperatures. In this model, the end points correspond to Brownian particles with different damping coefficients. Such analytical expression for the heat conductance covers its behavior from mesoscopic to very long one-dimensional quantum chains and validates the ballistic nature of the heat transport in the latter example. This implies the absence of the Fourier law for classical and quantum harmonic chains. We also provide a thorough analysis of the normal modes of system which helps us to satisfactorily interpret these results