Quantum Brownian motion and \\ its conflict with the second law
Theo M. Nieuwenhuizen
The Brownian motion of a harmonically bound quantum particle and coupled to a harmonic quantum bath is exactly solvable. At low enough temperatures the stationary state is non-Gibbsian due to an entanglement with the bath. This happens when a cloud of bath modes around the particle is formed. Equilibrium thermodynamics for particle plus bath together, does not imply standard thermodynamics for the particle itself at low $T$. Various formulations of the second law are then invalid. First, the Clausius inequality can be violated. Second, when the width of the confining potential is suddenly changed, there occurs a relaxation to equilibrium during which the rate of entropy production is partly negative. Third, for non-adiabatic changes of system parameters the rate of energy dissipation can be negative, and, out of equilibrium, cyclic processes are possible which extract work from the bath. Conditions are put forward under which perpetuum mobile of the second kind, having several work extraction cycles, enter the realm of condensed matter physics.
aipcheck.tex
%
% $Id: aipcheck.tex,v 1.7 2002/01/11 19:50:46 frank Exp $
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Testing for potential problems with this class
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newif\ifproblem
\newif\iftimesok

\makeatletter
\def\IfStandaloneCheck{\def\next{aipcheck}
  \edef\currjob{\jobname}
  \edef\next{\meaning\next}
  \edef\currjob{\meaning\currjob}
  \ifx\currjob\next
    \expandafter\@firstoftwo
  \else
    \expandafter\@secondoftwo
  \fi
}
\makeatother

\typeout{***********************************************}
\typeout{*}
\typeout{* Testing if all files required for the aipproc}
\typeout{* class are available ...}
\typeout{*}
\typeout{***********************************************}

\typeout{*}
\typeout{* Looking for LaTeX2e ... }
\ifx\documentclass\undefined
 \typeout{*}
 \typeout{* Sorry this is a fatal error:}
 \typeout{*}
 \typeout{* The aipproc class can only be used with LaTeX2e which is}
 \typeout{* the standard LaTeX since 1994!}
 \typeout{*}
 \typeout{* Please make sure that your version of LaTeX is up-to-date}
 \typeout{* before attempting to use this class.}
 \typeout{*}
 \expandafter\stop
\else
 \typeout{* ... ok }
\fi


\def\next#1/#2/#3\next{#1#2}
\typeout{*}
\typeout{* Testing that LaTeX2e is not too old ... }
\ifnum\expandafter\next\fmtversion\next<199612 \relax
 \typeout{* ... what a vintage! }
 \typeout{*}
 \typeout{* Sorry this is a fatal error:}
 \typeout{*}
 \typeout{* The aipproc class can only be used with a recent version}
 \typeout{* of LaTeX2e. Your version is dated \fmtversion\space --- but}
 \typeout{* at least the 1996/12/01 version is required!}
 \typeout{*}
 \typeout{* Please make sure that your version of LaTeX is up-to-date}
 \typeout{* before attempting to use this class.}
 \typeout{*}
 \expandafter\stop
\else
 \ifnum\expandafter\next\fmtversion\next<199806 \relax
   \typeout{* ... probably ok }
   \typeout{*}
   \typeout{* Your version of LaTeX2e is quite old --- the aipproc class}
   \typeout{* hasn't been tested with your release.}
   \typeout{*}
   \typeout{* We believe that it will probably work, but if you encounter}
   \typeout{* problems you will need upgrade your installation.}
   \typeout{*}
   \typein{* Type <return> to continue ...}
   \problemtrue
 \else
   \typeout{* ... ok }
 \fi
\fi


\typeout{*}
\typeout{* Looking for aipproc.cls ... }
\IfFileExists{aipproc.cls}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* Before you can use the aipproc class you have to unpack}
     \typeout{* it from the documented source.}
     \typeout{*}
     \typeout{* Run LaTeX on the file 'aipproc.ins', e.g.,}
     \typeout{*}
     \typeout{* \space\space latex aipproc.ins}
     \typeout{*}
     \typeout{* or whatever is necessary on your installation to process}
     \typeout{* a file with LaTeX. This should unpack a number of files for you:}
     \typeout{*}
     \typeout{* aipproc.cls \space and \space aip-*.clo}
     \typeout{*}
     \typeout{* After that retry processing this guide.}
     \typeout{*}
     \stop
}

\typeout{*}
\typeout{* Looking for aipxfm.sty ... }
\IfFileExists{aipxfm.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The aipxfm.sty file which is part of the aipproc distribution}
     \typeout{* must be installed in a directory which is searched by LaTeX.}
     \typeout{*}
     \typeout{* Please install this file and retry.}
     \typeout{*}
     \stop
}

\typeout{*}
\typeout{* Looking for aip-8s.clo ... }
\IfFileExists{aip-8s.clo}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The aip-8s.clo file which is part of the aipproc distribution}
     \typeout{* must be installed in a directory which is searched by LaTeX.}
     \typeout{*}
     \typeout{* Please install this file and retry.}
     \typeout{*}
     \stop
}

\typeout{*}
\typeout{* Looking for aip-8d.clo ... }
\IfFileExists{aip-8d.clo}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The aip-8d.clo file which is part of the aipproc distribution}
     \typeout{* must be installed in a directory which is searched by LaTeX.}
     \typeout{*}
     \typeout{* Please install this file and retry.}
     \typeout{*}
     \stop
}


\typeout{*}
\typeout{* Looking for aip-6s.clo ... }
\IfFileExists{aip-6s.clo}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The aip-6s.clo file which is part of the aipproc distribution}
     \typeout{* must be installed in a directory which is searched by LaTeX.}
     \typeout{*}
     \typeout{* Please install this file and retry.}
     \typeout{*}
     \stop
}


\typeout{*}
\typeout{* Looking for aip-arlo.clo ... }
\IfFileExists{aip-arlo.clo}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The aip-arlo.clo file which is part of the aipproc distribution}
     \typeout{* must be installed in a directory which is searched by LaTeX.}
     \typeout{*}
     \typeout{* Please install this file and retry.}
     \typeout{*}
     \stop
}

\typeout{*}
\typeout{* Looking for fixltx2e.sty ... }
\IfFileExists{fixltx2e.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found, trying fix2col.sty instead ... }
     \typeout{*}
     \IfFileExists{fix2col.sty}
	 {
	  \typeout{* ... ok }
	 }
	 {
	  \typeout{* ... not found! }
	  \typeout{*}
	  \typeout{* Sorry this is a fatal error:}
	  \typeout{*}
	  \typeout{* Your LaTeX distribution contains neither fixltx2e.sty}
	  \typeout{* nor fix2col.sty.}
	  \typeout{*}
	  \typeout{* This means that it is either too old or incompletely}
	  \typeout{* installed.}
	  \typeout{*}
	  \typeout{* fixltx2e.sty is part of the standard LaTeX distribution}
	  \typeout{* since 1999; fix2col.sty is an earlier version of this}
	  \typeout{* package.}
	  \typeout{*}
	  \typeout{* Best solution is to get the latest LaTeX distribution.}
	  \typeout{* If this is impossible for you, download fix2col.sty.}
	  \typeout{* You can get this software from a CTAN host.}
          \typeout{* Refer to http://www.ctan.org and search for "fix2col".}
	  \typeout{*}
	  \typeout{* After you have updated your LaTeX distribution}
	  \typeout{* retry processing this guide.}
	  \stop
     }
}

\typeout{*}
\typeout{* Looking for fontenc.sty ... }
\IfFileExists{fontenc.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The fontenc package, which is part of standard LaTeX}
     \typeout{* (base distribution) has to be installed at the site to}
     \typeout{* run the aipproc class.}
     \typeout{*}
     \typeout{* The fact that it cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* improperly.}
     \typeout{*}
     \typeout{* Please make sure that your version of LaTeX is okay}
     \typeout{* before attempting to use this class. The LaTeX distribution}
     \typeout{* contains the file "ltxcheck.tex" which can be used to}
     \typeout{* test the basic functionality and integrity of your installation.}
     \typeout{*}
     \stop
    }

\typeout{*}
\typeout{* Looking for calc.sty ... }
\IfFileExists{calc.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Sorry this is a fatal error:}
     \typeout{*}
     \typeout{* The calc package, which is part of standard LaTeX}
     \typeout{* (tool distribution) has to be installed at the site}
     \typeout{* to run the aipproc class.}
     \typeout{*}
     \typeout{* The fact that it cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts.}
     \typeout{*}
     \typeout{* Please make sure that the tools distribution of LaTeX}
     \typeout{* is installed before attempting to use this class.}
     \typeout{*}
     \typeout{* (You might be able to get calc.sty separately for your}
     \typeout{* installation if you are unable to upgrade to a recent}
     \typeout{* distribution for some reason.)}
     \typeout{*}
     \stop
    }

\typeout{*}
\typeout{* Looking for varioref.sty ... }
\IfFileExists{varioref.sty}
    {
     \typeout{* ... ok }
     \gdef\variorefoptionifavailable{varioref,}
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Problem detected:}
     \typeout{*}
     \typeout{* The varioref package, which is part of standard LaTeX}
     \typeout{* (tool distribution) is not installed at this site.}
     \typeout{*}
     \typeout{* The fact that it cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts.}
     \typeout{*}
     \typeout{* You can use the aipproc class without this package but }
     \typeout{* you cannot make use of the options "varioref" or "nonvarioref".}
     \typeout{*}
     \typeout{* Please also note that the aipguide.tex documentation}
     \typeout{* normally uses the "varioref" option to show its}
     \typeout{* effects (which  will now fail).}
     \typeout{*}
     \typein{* Type <return> to continue ...}
     \problemtrue
     \gdef\variorefoptionifavailable{}
     \let\vpageref\pageref
     \let\vref\ref
    }

\typeout{*}
\typeout{* Looking for times.sty ... }
\IfFileExists{times.sty}
    {
     \begingroup
% load times and forget it immediately again
       \RequirePackage{times}
       \global\expandafter\let\csname ver@times.sty\endcsname\relax    
       \long\def\next{ptm}
       \ifx\rmdefault\next
         \typeout{* ... ok }
         \gdef\psnfssproblemoption{}
         \endgroup
         \timesoktrue
       \else
         \endgroup
     \typeout{* ... obsolete! }
     \typeout{*}
     \typeout{* Serious problem detected:}
     \typeout{*}
     \typeout{* The times package, which is part of standard LaTeX}
     \typeout{* (psnfss distribution) is obsolete at this site.}
     \typeout{*}
     \typeout{* The fact that it contains incorrect code either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts with old files remaining!}
     \typeout{*}
     \typeout{* You can use the aipproc class without this package but}
     \typeout{* you have to specify the option "cmfonts" which result in}
     \typeout{* documents which are not conforming to the AIP layout specification!}
     \typeout{*}
     \typeout{* You can also try using the class in the following way:}
     \typeout{*}
     \typeout{* \space\space \string\documentclass[cmfonts]{aipproc}}
     \typeout{* \space\space \string\usepackage{times}}
     \typeout{* \space\space ...}
     \typeout{*}
     \typeout{* With luck this will result in Times Roman output but chances}
     \typeout{* are that you will get a larger number of error messages in}
     \typeout{* which case you have to remove the \string\usepackage declaration.}
     \typeout{*}
     \typein{* Type <return> to continue ...}
          \problemtrue
          \gdef\psnfssproblemoption{cmfonts}
          \def\textdegree{$^\circ$}            % used below but now
                                               % not setup
       \fi
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Serious problem detected:}
     \typeout{*}
     \typeout{* The times package, which is part of standard LaTeX}
     \typeout{* (psnfss distribution) can not be found.}
     \typeout{*}
     \typeout{* The fact that this package cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts!}
     \typeout{*}
     \typeout{* You can use the aipproc class without this package but }
     \typeout{* you have to specify the option "cmfonts" which result in}
     \typeout{* documents which are not conforming to the AIP layout specification!}
     \typeout{*}
     \typein{* Type <return> to continue ...}
     \problemtrue
     \gdef\psnfssproblemoption{cmfonts,}
    }

\iftimesok % don't bother testing other font options if times already
           % bad

\typeout{*}
\typeout{* Looking for t1ptm.fd or T1ptm.fd ... }
\IfFileExists{t1ptm.fd}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found, trying T1ptm.fd ... }
     \IfFileExists{T1ptm.fd}
          {
           \typeout{* ... ok }
          }
          {
           \typeout{* ... not found}
           \typeout{* Serious problem detected:}
           \typeout{*}
           \typeout{* The times package, which is part of standard LaTeX}
           \typeout{* (psnfss distribution) is available but the corresponding}
           \typeout{* .fd file (defining how to load Times Roman) is missing.}
           \typeout{*}
           \typeout{* The fact that this package is only partially installed}
           \typeout{* means that you LaTeX installation is unable to use Times}
           \typeout{* Roman fonts!}
           \typeout{*}
           \typeout{* You can use the aipproc class without this package but }
           \typeout{* you have to specify the option "cmfonts" which result in}
           \typeout{* documents which are not conforming to the AIP layout}
           \typeout{* specification!}
           \typeout{*}
           \typein{* Type <return> to continue ...}
           \problemtrue
           \timesokfalse
           \gdef\psnfssproblemoption{cmfonts,}
          }
    }

\fi


\newcommand\CheckFDFile[3]{%
  \typeout{*}
  \typeout{* Looking for #1#3.fd or #2#3.fd ... }
  \IfFileExists{#1#3.fd}
    {
     \typeout{* ... ok }
    }
    {
     \IfFileExists{#2#3.fd}
      {
       \typeout{* ... ok }
      }
      {\problemtrue
       \typeout{* ... not found! }
      }
    }
}

\iftimesok % don't bother testing other font options if Times already bad

%\CheckFDFile{ot1}{OT1}{ot1ztmcm}
%\CheckFDFile{oml}{OML}{omlztmcm}
%\CheckFDFile{oms}{OMS}{omsztmcm}
%\CheckFDFile{omx}{OMX}{omxztmcm}

\typeout{*}
\typeout{* Looking for mathptm.sty ... }
\IfFileExists{mathptm.sty}
    {
     \typeout{* ... ok }
     \CheckFDFile{ot1}{OT1}{ptmcm}
     \CheckFDFile{oml}{OML}{ptmcm}
     \CheckFDFile{oms}{OMS}{pzccm}
     \CheckFDFile{omx}{OMX}{psycm}
     \ifproblem
      \typeout{*}
      \typeout{* Problem detected:}
      \typeout{*}
      \typeout{* The mathptm package, which is part of standard LaTeX}
      \typeout{* (psnfss distribution) was found but some or all of its}
      \typeout{* support files describing which fonts to load are missing!}
      \typeout{*}
      \typeout{*}
      \typeout{* The fact that this package is only partially installed}
      \typeout{* means that the mathptm package cannot be used!}
      \typeout{*}
      \typeout{* You can use the aipproc class without this package but }
      \typeout{* you have to specify the option "nomathfonts" so that}
      \typeout{* math formulas will be typeset using Computer Modern.}
      \typeout{*}
      \typein{* Type <return> to continue ...}
      \problemtrue
      \gdef\psnfssproblemoption{nomathfonts,}
     \else
      \typeout{*}
      \typeout{* Looking for mathptmx.sty ... }
      \IfFileExists{mathptmx.sty}
       {
        \typeout{* ... ok }
        \CheckFDFile{ot1}{OT1}{ztmcm}
        \CheckFDFile{oml}{OML}{ztmcm}
        \CheckFDFile{oms}{OMS}{ztmcm}
        \CheckFDFile{omx}{OMX}{ztmcm}
        \ifproblem
	  \typeout{*}
	  \typeout{* Problem detected:}
	  \typeout{*}
	  \typeout{* The mathptmx package, which is part of standard LaTeX}
	  \typeout{* (psnfss distribution) was found but some or all of its}
	  \typeout{* support files describing which fonts to load are missing!}
	  \typeout{*}
	  \typeout{*}
	  \typeout{* The fact that this package is only partially installed}
	  \typeout{* means that the mathptmx package cannot be used!}
	  \typeout{*}
	  \typeout{* You can use the aipproc class without this package but }
	  \typeout{* you have to specify the option "mathptm" (no x) so that}
	  \typeout{* math formulas use the older version with upright greek letters.}
	  \typeout{*}
	  \typein{* Type <return> to continue ...}
	  \problemtrue
	  \gdef\psnfssproblemoption{mathptm,}
        \fi
       }
       {
	\typeout{* ... not found! }
	\typeout{*}
	\typeout{* Problem detected:}
	\typeout{*}
	\typeout{* The mathptmx package, which is part of standard LaTeX}
	\typeout{* (psnfss distribution) can not be found.}
	\typeout{*}
	\typeout{* This is unfortunate but not a disaster as the older}
	\typeout{* version of the package "mathptm" (no x) seems to exist.}
	\typeout{*}
	\typeout{* You can use the aipproc class without this package but }
	\typeout{* you have to specify the option "mathptm" so that}
	\typeout{* math formulas use the older version with upright greek letters.}
	\typeout{*}
	\typein{* Type <return> to continue ...}
	\problemtrue
	\gdef\psnfssproblemoption{mathptm,}
       }
      \fi
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Problem detected:}
     \typeout{*}
     \typeout{* The mathptm package, which is part of standard LaTeX}
     \typeout{* (psnfss distribution) can not be found.}
     \typeout{*}
     \typeout{* The fact that this package cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts!}
     \typeout{*}
     \typeout{* You can use the aipproc class without this package but }
     \typeout{* you have to specify the option "nomathfonts" so that}
     \typeout{* math formulas will be typeset using Computer Modern.}
     \typeout{*}
     \typein{* Type <return> to continue ...}
     \problemtrue
     \gdef\psnfssproblemoption{nomathfonts,}
    }

\typeout{*}
\typeout{* Looking for mathtime.sty ... }
\IfFileExists{mathtime.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* The mathime package can not be found.}
     \typeout{*}
     \typeout{* This is not a serious problem because this package is}
     \typeout{* only of interest if you own the commerical MathTime fonts.}
     \typeout{*}
     \typeout{* You can use the aipproc class without this package but }
     \typeout{* you cannot use the "mathtime" option of the class.}
     \typeout{*}
     \typein{* Type <return> to continue ...}
     \problemtrue
    }
\else
\fi % iftimesok

\typeout{*}
\typeout{* Looking for graphicx.sty ... }
\IfFileExists{graphicx.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Problem detected:}
     \typeout{*}
     \typeout{* The graphics package, which is part of standard LaTeX}
     \typeout{* (graphics distribution) can not be found.}
     \typeout{*}
     \typeout{* The fact that this package cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts!}
     \typeout{*}
     \typeout{* You can use the aipproc class without this package but }
     \typeout{* you cannot use commands like \protect\includegraphics
                or \protect\resizebox}
     \typeout{* in this case.}
     \typeout{*}
     \typeout{* Please note that you will get a further error message below}
     \typeout{* about: "graphicx.sty not found" because the class will try}
     \typeout{* to load this package! Type return in response to that error.}
     \typeout{*}
     \typeout{* As a result the illustrations in aipguide will look strange.}
     \typeout{*}
     \typein{* Type <return> to continue ...}

     \gdef\resizebox##1##2{}
     \gdef\includegraphics{\textbf{graphics package missing:}}
     \problemtrue
    }

\typeout{*}
\typeout{* Looking for textcomp.sty ... }
\IfFileExists{textcomp.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Problem detected:}
     \typeout{*}
     \typeout{* The textcomp package, which is part of standard LaTeX}
     \typeout{* (base distribution) can not be found.}
     \typeout{*}
     \typeout{* The fact that this package cannot be found either means that}
     \typeout{* this LaTeX release is too old or that it was installed}
     \typeout{* only in parts!}
     \typeout{*}
     \typeout{* You can use the aipproc class without this package but }
     \typeout{* you will always get the error: "textcomp.sty not found"}
     \typeout{* because the class will try to load this package!}
     \typeout{* Type return in response to that error.}
     \typeout{*}
     \typein{* Type <return> to continue ...}

     \def\textdegree{$^\circ$}         % used below but now
                                       % not set up
     \problemtrue
    }


\typeout{*}
\typeout{* Looking for url.sty ... }
\IfFileExists{url.sty}
    {
     \typeout{* ... ok }
    }
    {
     \typeout{* ... not found! }
     \typeout{*}
     \typeout{* Problem detected:}
     \typeout{*}
     \typeout{* The url package, which should be part of a good LaTeX}
     \typeout{* distribution, can not be found.}
     \typeout{*}
     \typeout{* Without this package you will not be able to use the \string\url}
     \typeout{* command. Try to download this package from a CTAN  host.}
     \typeout{* Refer to http://www.ctan.org and search for "url".}
     \typeout{*}
     \typein{* Type <return> to continue ...}

     \problemtrue
    }


\makeatletter

\typeout{*}
\typeout{* Looking for natbib.sty ... }
\IfFileExists{natbib.sty}
    {
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\begin{document}

\title{Quantum Brownian motion and \\ its conflict with 
the second law}


\author{Theo M. Nieuwenhuizen}
{address={Institute for Theoretical Physics, 
% University of Amsterdam,
Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands}}

\author{Armen E. Allahverdyan}
{address={Yerevan Physics Institute,
Alikhanian Brothers St. 2, Yerevan 375036, Armenia }}
\date{\today}



\begin{abstract}

The Brownian motion of a harmonically bound quantum particle 
and coupled to a harmonic quantum bath is exactly solvable. 
At low enough temperatures the stationary state is non-Gibbsian 
due to an entanglement with the bath. 
This happens when a cloud of bath modes
around the particle is formed.
Equilibrium thermodynamics for particle plus bath together, 
does not imply standard thermodynamics for the particle itself at low $T$.
Various formulations of the second law are then invalid. 
First, the Clausius inequality can be violated. 
Second, when the width of the confining potential is suddenly changed, 
there occurs a relaxation to equilibrium during which
the rate of entropy production is partly negative. 
Third, for non-adiabatic changes of system parameters the rate of 
energy dissipation can be negative, and, out of equilibrium, cyclic
processes are possible which extract work from the bath.
Conditions are put forward under which perpetuum mobile of the 
second kind, having several work extraction cycles, 
enter the realm of condensed matter physics.

 \end{abstract}

\maketitle

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{\bf Introduction.}
There are not two fundamental theories of nature, quantum mechanics and 
thermodynamics. There is only one: quantum mechanics, while
thermodynamics must emerge from it.
The universal character of equilibrium thermodynamics led to the 
general expectation that in one way or another, thermodynamics
will apply to the full quantum domain~\cite{landau}. 
Few people have taken the painful road to check this emergence,
yet this is what we have set out to do. Here we discuss the results
for quantum Brownian motion that have been presented
~\cite{ANQBMprl,NAlinw} and were discussed in the scientific literature
\cite{AIP}.

Brownian motion has numerous applications in condensed matter physics
\cite{klim,weiss,risken,leggett}, atomic physics
\cite{klim}, quantum optics and chemistry \cite{gardiner}. 
Some realizations involve weak coupling with the thermal 
bath \cite{gardiner}. However, there are well-known experimental 
situations, which are essentially far from the weak-coupling regime.
Here standard thermodynamics may not apply.
The main example of this is the case of weak links between superconductive
regions,  the so-called Josephson junctions, in their overdamped regime 
\cite{van,likho}, where the relevant ranges of parameters 
were achieved already twenty years ago. Even in quantum optics, which has
often been satisfactorily described by weak-coupling  theories 
\cite{gardiner}, there are recent experiments showing the 
necessity for moderate and strong coupling approaches~ \cite{oe}. 


{\bf The Hamiltonian.} We consider an `ideal gas' of non-interacting
harmonic oscillators coupled to a bath.
For the total Hamiltonian $\H_{\rm tot}=\H+\H_B+\H_I$ 
we thus assume~\cite{weiss}
\BEA
\label{hamiltonian}
\H=\frac{p^2}{2m}+\half a x^2, \quad \H_B=\sum_{i}\left [
\frac{p_i^2}{2m_i}+\frac{m_i\omega_i^2}{2}x_i^2\right], \quad 
\H_I=\sum_i\left[-c_ix_ix+\frac{c_i^2}{2m_i\omega_i^2}\,x^2\right ],
\EEA
with $\H$ describing the particle, $\H_B$ the bath, 
and $\H_I$ the interaction.

The bath is assumed to have uniformly spaced modes 
$ \omega_i=i\Delta,\qquad i=1,2,3,\cdots$.
The thermodynamic limit for the bath is taken  by sending 
$\Delta\to 0$, which induces relaxational behavior.
For the couplings we choose the quasi-Ohmic Drude-Ullersma 
spectrum, where \cite{weiss}
$J(\omega)=\half{\pi}\sum_i ({c_i^2}/{m_i\omega_i})
\delta(\omega-\omega_i)=
{\gamma \omega\Gamma^2}/(\omega^2+\Gamma^2)$,
where $\gamma$ is the damping constant. 
We shall assume that the Debye cutoff $\Gamma$ is large.

We describe an ensemble of closed total systems with conserved energy, 
except for the periods when work is done on it by externally changing 
$m$ or $a$. We take as initial density matrix the Gibbs distribution
$\exp(-\beta \H_{\rm tot})/Z_{\rm tot}$, with $Z_{\rm tot}$ the
partition sum.

{\bf Equilibrium state.}
In Gibbsian equilibrium of the total system the
free energy reads
$F_{\rm tot}(T,\gamma)= F_B(T,\gamma=0)
+F_p(a,\gamma,\Gamma,m,T)$.
The free energy of the bath itself,
\BEA\label{Ftotal2=} F_B(T,\gamma=0)=\frac{T}{\Delta}
 \int_0^\infty\d\omega\ln(1-e^{-\beta\,\hbarit\omega})
= -\frac{\pi^2T^2}{6\,\hbarit\Delta}, \label{comrad}\EEA
is of order $1/\Delta$, showing the extensivity of the bath.
The Brownian particle adds to this 
\BEA
\label{artush}
 F_p=T\left[\ln\Gamma\left(\frac{\beta\,\hbarit\Gamma}{2\pi}\right)
-\ln\Gamma\left(\frac{\beta\,\hbarit \omega_1}{2\pi}\right)
-\ln\Gamma\left(\frac{\beta\,\hbarit \omega_2}{2\pi}\right)-
\ln\Gamma\left(\frac{\beta\,\hbarit \omega_3}{2\pi}\right)
-\ln\frac{\beta\,\hbarit\omega_0}{(2\pi)^2}\right].
\EEA
It contains three characteristic, temperature independent of frequencies.
For small $\gamma $ (underdamping) they read:
$\omega _{1,2}= \pm i \omega _0 +{\gamma}/{2m}$,
$\omega _{3}=\Gamma  - {\gamma}/{m}$.
On the other hand, for strong damping $\gamma^2\gg am$,
$\omega _{1}={a}/{\gamma}$,
$\omega_2=({\gamma}/{m})(1-{am}/{\gamma^2})$,
$\omega _{3}= \Gamma-{\gamma}/{m}$.

{\it Effective temperatures.}
We shall now study two objects,  $T_x=a\langle x^2\rangle$
and $ T_p=\langle p^2\rangle/m$,  that would in classical
equilibrium be equal to $T$ and which we shall interpret below
as effective temperatures. 
As in the classical situation, it holds that
$T_x=2a{\partial F_p}/{\partial a}$, $T_p
=-2m{\partial F_p}/{\partial m}$.
For $\gamma\to 0$ one gets the weak-coupling result known from
textbooks, 
$U\equiv\half T_x+\half T_p=T_x=T_p={\half\,\hbarit \omega _0}
\coth\half{\beta\,\hbarit\omega _0}$, with  $\omega_0=\sqrt{a/m}$. 
At large $T$ one gets
\BEA\label{TxTplargeT}
T_x=T+\frac{\beta\,\hbarit^2a}{12m},\quad
T_p=T+\frac{\beta\,\hbarit^2(a+\gamma\,\Gamma)}{12m},\quad
F_p= T\ln\beta\,\hbarit\sqrt{\frac{a}{m}}+
\frac{\beta\,\hbarit^2(a+\gamma\,\Gamma)}{24m}.
\EEA
At low $T$ and for strong damping one has
\BEA
T_x=\frac{\,\hbarit a}{\pi\gamma}\ln\frac{\gamma^2}{am}
,\quad
T_p=\frac{\,\hbarit \gamma }{\pi m}\ln\frac{\Gamma m}{\gamma}+
\frac{\,\hbarit a}{\pi \gamma} %+{\cal O}(T^4)
,\quad
F_p=\frac{\,\hbarit \gamma}{2\pi m}
\ln\frac{e\Gamma m}{\gamma}+
\frac{\,\hbarit a}{2\pi \gamma}\ln\frac{\gamma^2}{a m}.
\label{mega3}
\EEA
The fact that $T_p\ge T_x>0$ at $T=0$ is related to the quantum
nature of the problem. 

{\bf Generalized thermodynamic formulation.}
The Wigner function has a quasi-Gibbsian expression, 
since there occur two  temperature-like variables,
\BEQ \label{Wpx=}
W(p,x)=W_p(p)W_x(x)=\frac{e^{-p^2/2mT_p}}{\sqrt{2\pi mT_p}}\,\,
\frac{e^{-ax^2/2T_x}}{\sqrt{2\pi T_x/a}}.\EEQ
The Boltzmann entropy of momentum and coordinate is
\BEA 
\label{kow3}
&&S_p=-\int \d p W_p(p)\ln[W_p(p)\,\sqrt{\,\hbarit}\,]=
\half \ln \frac{emT_p}{\,\hbarit},
\quad S_x= \half \ln \frac{eT_x}{\,\hbarit a},
 \EEA
while the von Neumann entropy reads~\cite{weiss}
$S_{vN} =(v+\half)\ln(v+\half) -(v-\half)\ln(v-\half)$, with
$v={\Delta p\,\Delta x}/{\,\hbarit}=
\sqrt{{\langle p^2\rangle\langle x^2\rangle}}/{\,\hbarit}=
\sqrt{{mT_pT_x}}/\,\hbarit \sqrt{a}$.
The first terms in its large $v$-expansion,
$S_{vN} =\ln v+1$, coincide with the total Boltzmann entropy
$S_B=S_p+S_x$.




{\it Internal energy and interaction energy.}
The energy of the central particle reads
$U=\langle\H\rangle=\half T_p+\half T_x$.
The energy of the cloud of bath modes 
that surround the particle is   
$ U_{\rm int}=U_{\rm tot}-U_B(\gamma=0)-U=U_p-U=
\Gamma {\p F_p}/{\p\Gamma}$.
At high temperatures one has 
$ U=T+{\beta\,\hbarit^2}(2a+\gamma\,\Gamma)/24m$, 
$U_{\rm int}={\beta\,\hbarit^2\gamma\,\Gamma}/{24m}$.
Since the energy of the cloud involves $\,\hbarit$,  it is a quantum effect.
At low temperatures one has, if $\Gamma$ is large,
$U_{\rm int}={\,\hbarit\gamma}/{2\pi m}+{\cal O}(T^4)$.

{\it Implementation of the first law.}
Given the Hamiltonian $\H(p,x)$ of the subsystem, its
energy is $U=\langle \H\rangle=\int \d p\d x \H(p,x)W(p,x)$,
where $W$ is the Wigner function of the subsystem.
When we change a system parameter, the energy changes as
$\d U = \d\int \H W = \int\H \, \d W+\int W\, \d \H
\equiv \dbarrm {\cal Q}+\dbarrm {\cal W}$. 
The first term  represents the
variation due to the statistical redistribution of the 
phase space, which we identify with the change in heat 
$\,\dbarrm {\cal Q}$. 
The last term results from the change in the Hamiltonian,
so it is a mechanical, non-statistical object, which
we associate with the work $\,\dbarrm {\cal W}$ 
done by external sources. 



{\it Generalized free energy and the second law.}
The definition of the effective temperatures 
admits a thermodynamical interpretation.
The free energy $F$ for a two-temperature
system is  defined as ~\cite{1} 
$F=U-T_pS_p-T_xS_x=-\half T_p\ln m T_p-\half T_x\ln({T_x}/{a})$.
For adiabatic changes in $m$ or $a$ one has 
$\d F=- S_x\d T_x- S_p\d T_p+\dbarrm {\cal W}$,
with work
$\dbarrm {\cal W}
=\langle{\partial \H}/{\partial m}\rangle\d m
+\langle{\partial \H}/{\partial a}\rangle\d a
=-T_p{\d m}/{2m}+T_x{\d a}/{2a}$.
Due to the first law this yields the 
second law for situations with two temperatures,
$\dbarrm {\cal Q}= T_p\d S_p+T_x\d S_x$,
 in close analogy with those proposed recently for 
nonequilibrium glassy systems \cite{1}
and black holes~\cite{Nbh}. 
Notice that $F$ pertains to the particle alone, and does not
satisfy Gibbsian thermodynamics,
while the Gibbsian $F_p$ of Eq. (\ref{artush}),
 relates to the whole equilibrium system, i.e., 
to the particle and the cloud of bath modes around it. 
There are many physical systems, such as a Josephson
junction strongly coupled to the electromagnetic field, 
where the natural object to study is $F$,  
relating to properties of the junction only.

{\it Violation of the Clausius inequality at low $T$.}
The Clausius inequality $\dbarrm {\cal Q}\le T\d S_{vN}$ is one of the
formulations of the second law. 
At $T=0$ it says that no heat can be taken from the bath, 
at best heat can go from the subsystem  to the  bath. 

When we change $a\to a+\d a$, $\dbarrm {\cal Q}$ is of order $-T^2\d a$, 
while $T\d S_{vN}$ is of order $-T\d a$. 
In the case $\d a >0$, where an amount of work 
$\dbarrm\, {\cal W}_{\rm rev}\sim \d a$ is done on the system, 
the Clausius relation is thus violated at low $T$.
Likewise one can consider the variation of the (effective) mass $m$.
Here one has $\dbarrm {\cal Q}
%=\half({\p T_p}/{\p m}+{\p T_x}/{\p m}+{T_p}/{m}){\d m}
=\,\hbarit \gamma\,\d m/{2\pi m^2}$.
Now there is a transfer of heat even
if the bath temperature is zero. Thus, violation of the Clausius inequality 
is even stronger in this case.
This situation with $\dbarrm\, {\cal W}_{\rm rev}<0$ corresponds to the work 
performed by the system on the environment. That the heat 
comes from the cloud of bath modes, is confirmed by the fact that 
$ \dbarrm {\cal Q}_{\rm rev}=-\d U_{\rm int}$.

{\it Violation of the Landauer bound for information erasure.}
A further aspect is the squeezing of phase space and entropy, relevant 
for computing in the quantum regime. We have shown that the so-called 
Landauer bound $-\dbarrm \Q\ge k T\,\ln 2$ for the minimal energy 
dispersion when erasing one bit of information is violated in a similar 
manner~\cite{ANQBMprl}. This connection arises because the Landauer 
bound is just based on the Clausius inequality. 




{\bf Dynamics from a non-equilibrium state.}
{\it  Energy oscillation at low $T$.} 
We consider the dynamical evolution of a system initially in 
equilibrium characterized by a spring constant $a_0$, which
at $t=0$ is instantaneously changed to $a_1=a$. 
These parameters are connected as $ a_0=(1-\alpha_0)a$ and 
we assume that $|\alpha_0|\ll 1$. 
For strong damping one has for the energy of the subsystem
$U(t)=U(\infty)+(\,\hbarit a\alpha_0/2\pi\gamma)f_1(at/\gamma)$.
The behavior of $f_1(\sigma)$ at different temperatures is presented in
Fig. 1a. For $\alpha_0>0$ it says that, after initially energy has been put
on the particle by the change of $a_0\to a>a_0$, this energy leaks away 
into the bath. At low $T$, however, too much leaks away, 
and a part has to come back.
This ``bouncing'' is familiar of the noise correlator, which behaves 
like $f_1$.

\begin{figure}
\includegraphics[width=6cm]{NApmfig4.eps}\hspace{1cm}
\includegraphics[width=6cm]{PeriodSpring.eps}
\caption{a) Normalized excess energy $f_1$ as a function 
of $at/\gamma$, 
for different values of the temperature. Upper curve: $T\to\infty$.
Middle curve: $T=\,\hbarit a/2\gamma$. 
Lower curve: at $T=0$ it is non-monotonous. \hspace{1cm}
b) Schematic plot of the cyclic changes in the spring constant,
where successive cycles are slower and slower.
$h$ characterizes the size of the change and $t$ denotes
the dimensionless time. The interval $-\infty<t<t_1$
marks the process that establishes the nonequilibrium state at
$t=t_1$. The picture shows three full cycles, in the intervals
$t_i<t<t_{i+1}$, ($i=1,2,3$). Their start and end points are indicated by
bullets.}
\label{fig1}
\end{figure}




{\it Entropy production.}
The rate of production of Boltzmann entropy is found after a lengthy
identification of its flux. The result can
be negative at moderate $T$~\cite{NAlinw}.
For $T=0$ and strong damping one has 
with $\sigma=at/\gamma$ and  $\sigma_{min}=0.879$:  
${\d _iS_x}/{\d t}\sim + \alpha_0^2(\sigma-\taum)(\sigma-\taum 
+\eps)$, where $\eps=am/\gamma^2$.
This is negative for $\taum-\eps<\sigma<\taum$, 
or $\Delta t=m/\gamma$, the characteristic timescale of the momentum.


{\it Smooth changes of the spring constant.}
Let us now consider the case where, 
starting from the equilibrium state $a(-\infty)=a$, the spring constant 
$a(t)=[1-\alpha(t)]a$, is slightly changed ($|\alpha(t)|\ll 1$) in a 
smooth way.
The rate of work added to the system is
${\d \W}/{\d t}={\d \W_{\rm rev}}/{\d t}+{\d \Pi}/{\d t}$,
with the adiabatic (`reversible') and dispersed contributions
\BEQ \frac{\d \W_{\rm rev}}{\d t}=
-\frac{\gamma\,\alpha'(\tau)}{4m}\,
\left[T_x+\frac{\,\hbarit a}{\pi\gamma }\alpha(\tau)C_x(0)\right], 
\quad \frac{\d \Pi}{\d t}= \frac{\,\hbarit a\,\alpha'(\tau)}{4\pi m}\,
\int_0^\infty\d \sigma\,\alpha'(\tau-\sigma) C_x(\sigma). \nn
\EEQ
where $\tau=\gamma\, t/2m$ and $C_x$ a function like $f_1$ of Fig. 1a.  
Integrating over the full change
(the whole region where $\alpha'\neq 0$)  one confirms
that the dispersion for a completed, cyclic change 
($\alpha_i=\alpha_f$) of system parameters is nonnegative,
the Thomson formulation of the second law~\cite{ANthomson}. 
It is also nonnegative for noncyclic ($\alpha_i\neq \alpha_f$) 
but completed changes. 

{\it Energy dispersion.}
If $\alpha(\tau)=\alpha_m h(\Omega t)$, with $\Omega$ a  slow 
rate of change, and $\gamma$ is large, expansion in $\Omega$ is 
possible. Omitting numerical factors one gets the structure
\BEA\label{dPIdt=}\frac{\d \Pi}{\d t}=
\,\hbarit\alpha_m^2\Omega^2\,h'\,
\left[\left(\frac{\gamma T}{\,\hbarit a}\right)^2h'
+\frac{\gamma\Omega}{a} h''-\frac{\gamma^2\Omega^2}{a^2}h'''\right]. \EEA
The full integral is always positive and for moderate $T$ the 
first term is dominant. However, for small $T$ the term
$h'h''$ may imply that ${\d \Pi}/{\d t}$ is negative.
This is a firm statement, also valid when starting in equilibrium,
since work is the energy added to the total system. 
It is the more surprising since the relevant domain the characteristic
timescale of the change of $a$,  
$1/\Omega <\,\hbarit^2/(\gamma T^2)$,  may exceed all other timescales,
including the quantum timescale $\,\hbarit/T$, so that applicability
of thermodynamics was to be expected.

 



{\it Perpetuum mobile with many work extraction cycles.}
The $h'h''$ term can cause extraction of work. 
Let there be $\N$ non-overlapping Gaussians, $h(x)=\exp(-\half x^2)$,
as depicted in Figure 1b. 
Cycles are the intervals $t_i<t<t_{i+1}$ ($i\ge 1$).
Each new cycle is slower than the previous one, $\Omega_{n+1}<\Omega_n$. 
One can make a total number $\N\sim 1/T$ of cycles with equal yield, where
$\N$ is parameterized by $v$, and total yield 
\BEQ \N=\frac{\,\hbarit a}{2\pi\gamma T}\int_{\beta\,\hbarit\Omega_\N/2\pi}
^{\beta\,\hbarit\Omega_1/2\pi}\frac{\d y y}{v+I_1y+I_3y^3},
\quad \W_{\rm tot}=
-\frac{\pi \amsq }{3}\,\frac{\gamma T^2}{\,\hbarit a}
\int_{\beta\,\hbarit\Omega_\N/2\pi}^{
\beta\,\hbarit\Omega_1/2\pi}\d y\frac{v\, y}{v+I_1y+I_3y^3}.
\EEQ 
with $I_1=\half\sqrt{\pi}$, $I_3=3\sqrt{\pi}/4$.
The minus sign of $\W_{\rm tot}$ indicates that
work is performed by the system on the environment. 
This is possible because,
in order to make the extraction cycles, one had to start from the
equilibrium state $\alpha(-\infty)=0$ and change $\alpha$ 
up to $\alpha(t_1)$. In this first part the  
 energy dispersion was 
$\Pi(t_1)=+{\amsq }{\,\hbarit\gamma}\Omega_1^2/24\pi a
\,\, > \,\left|\W_{\rm tot}\right|$.


{\bf Feasibility.} 
The harmonic oscillator can be interpreted as an LC circuit \cite{klim-rev}.
$x$ may correspond to the charge $Q$ on a capacitor, $1/a$ to its
capacitance $C$, $m$ to an inductance $L$, $p$ to a phase $\Phi$,
$\gamma $ to a resistance $R$, and $\eta (t)$ to a random 
electro-motoric force. In this setup there should be
nothing difficult in varying $L\sim m$ or $C\sim 1/a$.
We have proposed to test the violation of the Clausius inequality
in mesocopic linear circuits~\cite{NArlc}. This amounts to measuring
$\langle \delta Q^2\rangle$, $\langle \delta \Phi^2\rangle$ 
and the produced work. The conditions to do this experiment were
reached 20 years ago, and one of the quantities,
$\langle \delta Q^2\rangle$, was measured already and agrees perfectly
with the theoretical predictions ~\cite{w1,flabby}.




{\bf Has thermodynamics been violated?} 
 Let us recall that our results hold also
for $N$ non-interacting Brownian particles in a bath.
% those results have factors $N$ at appropriate places.
Our conclusion is that thermodynamics does not 
always work when, in the quantum regime, one considers 
the Brownian particle in its reduced Hilbert space,
thus  summing out the bath.
We should admit that at low enough $T$ 
the interaction energy between system and bath becomes
relevant, since the damping constant is fixed. 
% Even though not many people seemed aware of this problem,
In a very strict definition of thermodynamics one may
claim that there is no reason why thermodynamics had to apply.

For this reason our surprising findings of the breakdown of the Clausius 
inequality  can in principle be viewed as results outside the domain 
of applicability of thermodynamics.
However, the negativity of the Boltzmann entropy production sets 
in already at moderate $T$, more or less in the same domain where 
energy initially put on the particle starts to dissipate to the bath 
in a non-monotonic fashion, even in limit 
of large damping.

Our second effect in that regime, the presence of many work extraction 
cycles (``perpetuum mobile'') involves only the energy budget of the total
system, and should be more surprising. However, when starting in an
equilibrium state and counting all the energy, the total dispersion
is positive, as demanded by an exact theorem~\cite{ANthomson}.


These extraction cycles relied on the fact that 
the rate of energy dispersion by the total system can be negative.
This is a firm and unexpected statement about the total system,
that may have started in equilibrium.
Thus positivity of the rate of energy dispersion ~\cite{KondPrig} 
is not always a good formulation of the second law.

The new aspects of quantum 
Brownian motion arise from quantum entanglement: A complete description 
in terms of a wave function is possible only for a closed system;
subsystems are necessarily in a mixed state. Thus the quantum Gibbs 
distribution is not an adequate candidate for the description of the
quantum subsystem non-weakly interacting with its thermal bath. 
Connections of these findings with NMR physics~\cite{PAN},
mesoscopic work sources~\cite{ANthomson} and with
Maxwell's demon ~\cite{demon} are discussed elsewhere.




{\it Acknowledgments.}
We thank R. Balian and L.S. Suttorp for discussion.
 
\begin{thebibliography}{99}

\bibitem{landau} L.D. Landau and E.M. Lifshitz,
{\it Statistical Physics, Part 1}, (Pergamon, London, 1980).	


\bibitem{ANQBMprl} A.E. Allahverdyan and Th.M. Nieuwenhuizen,
{\it Extracting work from a single thermal bath in the quantum regime},
Phys. Rev. Lett. {\bf 85} (2000) 1799;
%\bibitem{ANLand}A.E. Allahverdyan and Th.M. Nieuwenhuizen, 
{\it Breakdown of the Landauer bound for information erasure 
in the quantum regime},
Phys. Rev. E {\bf 64}, 056117 (2001).

\bibitem{NAlinw} 
Th.M. Nieuwenhuizen and A.E. Allahverdyan,
{\it  Statistical thermodynamics of quantum Brownian motion: 
Construction of perpetuum mobile of the second kind}, 
Phys. Rev. E (2002); cond-mat/0011389. 

\bibitem{AIP}Philip F. Schewe and Ben Stein, 
{\it New frontiers of thermodynamics}, 
http://www.aip.org/enews/
physnews/2000/split/pnu494-1.htm;  Peter Weiss, 
{\it Breaking the law: Can quantum mechanics + thermodynamics 
= perpetual motion ?}, Sci. News 
(Washington D.C.){\bf 158} 234 (2000).


\bibitem{klim} Yu. L. Klimontovich,
{\it Statistical Theory of Open Systems}, (Kluwer, Amsterdam, 1997).

\bibitem{weiss} U. Weiss, {\it Quantum Dissipative Systems}, 
(World Scientific, Singapore, 1993).


\bibitem{risken}H. Risken, {\it The Fokker-Planck Equation}, 
(Springer, Berlin, 1984).


\bibitem{leggett} 
A.O. Caldeira and A.J. Leggett,  
%{\it Quantum tunneling in a dissipative system}, 
Ann. Phys.  {\bf 149}, 374, (1983).


\bibitem{gardiner} C.W. Gardiner, {\it Quantum Noise}, 
(Springer-Verlag, 1991).

\bibitem{van}R.H. Koch, D.J. Van Harlingen and J. Clarke, 
%{\it Quantum-Noise Theory for the Resistively Shunted Josephson Junction},
Phys. Rev. Lett. {\bf 45}, 2132, (1980); 
%{\it Observation of Zero-Point Fluctuations in a Resistively 
%Shunted Josephson Tunnel Junction}, 
ibid {\bf 47}, 1216 (1981).

\bibitem{likho} K.K. Likharev, 
%{\it True quantum-mechanical macroscopic effects in weak
%superconductivity},  
Sov. Phys. Usp. {\bf 26}, 87 (1983).

\bibitem{oe}K. Suominen, B. M. Garraway, and S. Stenholm,
%{\it Wave-packet model for excitation by ultrashort pulses},
Phys. Rev. A {\bf 45} 3060 (1992).








\bibitem{1} Th.M. Nieuwenhuizen, 
{\it  Thermodynamic description of a dynamical glassy transition},
J. Phys. A {\bf 31}, L201, (1998);
{\it Thermodynamics of the glassy state: effective temperature 
as an additional system parameter}, Phys. Rev. Lett. {\bf 80} 5580 (1998);
{\it  Thermodynamic picture of the glassy state gained from exactly 
solvable models}, Phys. Rev. E {\bf 61} 267 (2000); 
{\it Formulation of thermodynamics for the glassy state: 
configurational energy as a modest source of energy},
J. Chem. Phys. {\bf 115} 8083 (2001);
% \bibitem{4}
 A.E. Allahverdyan and Th.M. Nieuwenhuizen,
{\it Entropy production, energy dissipation and violation of Onsager
relations in the steady adiabatic state}, Phys. Rev. E {\bf 62} 845 (2000).

\bibitem{Nbh} Th.M. Nieuwenhuizen, 
{\it Thermodynamics of black holes: an analogy with glasses},
Phys. Rev. Lett. {\bf 81} 2201 (1998).

\bibitem{ANthomson}  
A.E. Allahverdyan, R. Balian  and Th.M. Nieuwenhuizen,
{\it Thomson's formulation of the second law:
an exact theorem and limits of its validity}, these proceedings and
preprint 2002; 
A.E. Allahverdyan and Th.M. Nieuwenhuizen,
{\it A mathematical theorem as the basis for the second law: Thomson's
formulation applied to equilibrium}, Physica A {\bf 305},  542 (2002).


\bibitem{klim-rev}Yu. L. Klimontovich,
%{\it Fluctuation-dissipation relations. Role of the finiteness of the
%correlation time. Quantum generalization of Nyquist's formula},
 Sov. Phys. Usp. {\bf 30}, 154 (1987).

\bibitem{NArlc} A.E. Allahverdyan and Th.M. Nieuwenhuizen,
{\it On testing the violation of the Clausius inequality in nanoscale 
electric circuits}, Phys. Rev. B, to appear; cond-mat/0205156.

\bibitem{w1} A.N. Cleland, J.M. Schmidt and J. Clarke, 
% {\it Influence of the environment on the Coulomb blockade in 
% submicrometer normal-metal tunnel junctions},
Phys. Rev. B {\bf 45}, 2950, (1992).

\bibitem{flabby}
G.Y. Hu and R.F. O`Connell,
% {\it Charge fluctuations and zero-bias resistance in small-capacitance 
% tunnel junctions},  
Phys. Rev. B {\bf 46}, 14 219, (1992).



\bibitem{KondPrig} P. Kondopudi and I. Prigogine, 
{\it Modern thermodynamics: from heat engines to dissipative structures},
(Wiley, West Sussex, UK, 1998).

\bibitem{PAN} C. Pombo, A.E. Allahverdyan and Th.M. Nieuwenhuizen, 
{\it Bath generated work extraction in two-level systems}
these proceedings;  A.E. Allahverdyan and Th.M. Nieuwenhuizen, 
{\it Bath generated work extraction and inversion-free gain 
in two-level systems}, cond-mat/0201408.

\bibitem{demon} Th.M. Nieuwenhuizen and A.E. Allahverdyan, 
{\it Unmasking  Maxwell's Demon}, these proceedings.



\end{thebibliography}
\end{document}
cond-mat0208564.extracted_bib
\begin{thebibliography}{99}

\bibitem{landau} L.D. Landau and E.M. Lifshitz,
{\it Statistical Physics, Part 1}, (Pergamon, London, 1980).	


\bibitem{ANQBMprl} A.E. Allahverdyan and Th.M. Nieuwenhuizen,
{\it Extracting work from a single thermal bath in the quantum regime},
Phys. Rev. Lett. {\bf 85} (2000) 1799;
%\bibitem{ANLand}A.E. Allahverdyan and Th.M. Nieuwenhuizen, 
{\it Breakdown of the Landauer bound for information erasure 
in the quantum regime},
Phys. Rev. E {\bf 64}, 056117 (2001).

\bibitem{NAlinw} 
Th.M. Nieuwenhuizen and A.E. Allahverdyan,
{\it  Statistical thermodynamics of quantum Brownian motion: 
Construction of perpetuum mobile of the second kind}, 
Phys. Rev. E (2002); cond-mat/0011389. 

\bibitem{AIP}Philip F. Schewe and Ben Stein, 
{\it New frontiers of thermodynamics}, 
http://www.aip.org/enews/
physnews/2000/split/pnu494-1.htm;  Peter Weiss, 
{\it Breaking the law: Can quantum mechanics + thermodynamics 
= perpetual motion ?}, Sci. News 
(Washington D.C.){\bf 158} 234 (2000).


\bibitem{klim} Yu. L. Klimontovich,
{\it Statistical Theory of Open Systems}, (Kluwer, Amsterdam, 1997).

\bibitem{weiss} U. Weiss, {\it Quantum Dissipative Systems}, 
(World Scientific, Singapore, 1993).


\bibitem{risken}H. Risken, {\it The Fokker-Planck Equation}, 
(Springer, Berlin, 1984).


\bibitem{leggett} 
A.O. Caldeira and A.J. Leggett,  
%{\it Quantum tunneling in a dissipative system}, 
Ann. Phys.  {\bf 149}, 374, (1983).


\bibitem{gardiner} C.W. Gardiner, {\it Quantum Noise}, 
(Springer-Verlag, 1991).

\bibitem{van}R.H. Koch, D.J. Van Harlingen and J. Clarke, 
%{\it Quantum-Noise Theory for the Resistively Shunted Josephson Junction},
Phys. Rev. Lett. {\bf 45}, 2132, (1980); 
%{\it Observation of Zero-Point Fluctuations in a Resistively 
%Shunted Josephson Tunnel Junction}, 
ibid {\bf 47}, 1216 (1981).

\bibitem{likho} K.K. Likharev, 
%{\it True quantum-mechanical macroscopic effects in weak
%superconductivity},  
Sov. Phys. Usp. {\bf 26}, 87 (1983).

\bibitem{oe}K. Suominen, B. M. Garraway, and S. Stenholm,
%{\it Wave-packet model for excitation by ultrashort pulses},
Phys. Rev. A {\bf 45} 3060 (1992).








\bibitem{1} Th.M. Nieuwenhuizen, 
{\it  Thermodynamic description of a dynamical glassy transition},
J. Phys. A {\bf 31}, L201, (1998);
{\it Thermodynamics of the glassy state: effective temperature 
as an additional system parameter}, Phys. Rev. Lett. {\bf 80} 5580 (1998);
{\it  Thermodynamic picture of the glassy state gained from exactly 
solvable models}, Phys. Rev. E {\bf 61} 267 (2000); 
{\it Formulation of thermodynamics for the glassy state: 
configurational energy as a modest source of energy},
J. Chem. Phys. {\bf 115} 8083 (2001);
% \bibitem{4}
 A.E. Allahverdyan and Th.M. Nieuwenhuizen,
{\it Entropy production, energy dissipation and violation of Onsager
relations in the steady adiabatic state}, Phys. Rev. E {\bf 62} 845 (2000).

\bibitem{Nbh} Th.M. Nieuwenhuizen, 
{\it Thermodynamics of black holes: an analogy with glasses},
Phys. Rev. Lett. {\bf 81} 2201 (1998).

\bibitem{ANthomson}  
A.E. Allahverdyan, R. Balian  and Th.M. Nieuwenhuizen,
{\it Thomson's formulation of the second law:
an exact theorem and limits of its validity}, these proceedings and
preprint 2002; 
A.E. Allahverdyan and Th.M. Nieuwenhuizen,
{\it A mathematical theorem as the basis for the second law: Thomson's
formulation applied to equilibrium}, Physica A {\bf 305},  542 (2002).


\bibitem{klim-rev}Yu. L. Klimontovich,
%{\it Fluctuation-dissipation relations. Role of the finiteness of the
%correlation time. Quantum generalization of Nyquist's formula},
 Sov. Phys. Usp. {\bf 30}, 154 (1987).

\bibitem{NArlc} A.E. Allahverdyan and Th.M. Nieuwenhuizen,
{\it On testing the violation of the Clausius inequality in nanoscale 
electric circuits}, Phys. Rev. B, to appear; cond-mat/0205156.

\bibitem{w1} A.N. Cleland, J.M. Schmidt and J. Clarke, 
% {\it Influence of the environment on the Coulomb blockade in 
% submicrometer normal-metal tunnel junctions},
Phys. Rev. B {\bf 45}, 2950, (1992).

\bibitem{flabby}
G.Y. Hu and R.F. O`Connell,
% {\it Charge fluctuations and zero-bias resistance in small-capacitance 
% tunnel junctions},  
Phys. Rev. B {\bf 46}, 14 219, (1992).



\bibitem{KondPrig} P. Kondopudi and I. Prigogine, 
{\it Modern thermodynamics: from heat engines to dissipative structures},
(Wiley, West Sussex, UK, 1998).

\bibitem{PAN} C. Pombo, A.E. Allahverdyan and Th.M. Nieuwenhuizen, 
{\it Bath generated work extraction in two-level systems}
these proceedings;  A.E. Allahverdyan and Th.M. Nieuwenhuizen, 
{\it Bath generated work extraction and inversion-free gain 
in two-level systems}, cond-mat/0201408.

\bibitem{demon} Th.M. Nieuwenhuizen and A.E. Allahverdyan, 
{\it Unmasking  Maxwell's Demon}, these proceedings.



\end{thebibliography}