Nieuwenhuizen Th M, Allahverdyan A E
Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, The Netherlands.
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Sep;66(3 Pt 2A):036102. doi: 10.1103/PhysRevE.66.036102. Epub 2002 Sep 5.
The Brownian motion of a quantum particle in a harmonic confining potential and coupled to harmonic quantum thermal bath is exactly solvable. Though this system presents at high temperatures a pedagogic example to explain the laws of thermodynamics, it is shown that at low enough temperatures the stationary state is non-Gibbsian due to an entanglement with the bath. In physical terms, this happens when the cloud of bath modes around the particle starts to play a nontrivial role, namely, when the bath temperature T is smaller than the coupling energy. Indeed, equilibrium thermodynamics of the total system, particle plus bath, does not imply standard equilibrium thermodynamics for the particle itself at low T. Various formulations of the second law are found to be invalid at low T. First, the Clausius inequality can be violated, because heat can be extracted from the zero point energy of the cloud of bath modes. Second, when the width of the confining potential is suddenly changed, there occurs a relaxation to equilibrium during which the entropy production is partly negative. In this process the energy put on the particle does not relax monotonically, but oscillates between particle and bath, even in the limit of strong damping. Third, for nonadiabatic 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 mobility of the second kind, having one or several work extraction cycles, enter the realm of condensed matter physics. Fourth, it follows that the equivalence between different formulations of the second law (e.g., those by Clausius and Thomson) can be violated at low temperatures. These effects are the consequence of quantum entanglement in the presence of the slightly off-equilibrium nature of the thermal bath, and become important when the characteristic quantum time scale variant Planck's over 2pi /k(B)T is larger than or comparable to other time scales of the system. They show that there is no general consensus between standard thermodynamics and quantum mechanics. The known agreements occur only due to the weak coupling limit, which does not pertain to low temperatures. Experimental setups for testing the effects are discussed.
处于简谐限制势且与简谐量子热库耦合的量子粒子的布朗运动是完全可解的。尽管该系统在高温下为解释热力学定律提供了一个具有启发性的例子，但研究表明，在足够低的温度下，由于与热库的纠缠，稳态是非吉布斯态的。从物理角度来看，当粒子周围的热库模式云开始起重要作用时，即当热库温度(T)小于耦合能量时，就会出现这种情况。实际上，对于由粒子加 热库组成的总系统的平衡热力学，并不意味着在低温下粒子自身的标准平衡热力学。人们发现各种形式的第二定律在低温下都是无效的。首先，克劳修斯不等式可能会被违反，因为可以从热库模式云的零点能量中提取热量。其次，当限制势的宽度突然改变时，会出现向平衡的弛豫，在此过程中熵产生部分为负。在这个过程中，施加在粒子上的能量并非单调弛豫，而是在粒子和热库之间振荡，即使在强阻尼极限下也是如此。第三，对于系统参数的非绝热变化，能量耗散率可能为负，并且在非平衡状态下，可能存在从热库提取功的循环过程。提出了存在具有一个或几个功提取循环的第二类永动机进入凝聚态物理领域的条件。第四，由此可知，第二定律的不同表述（例如克劳修斯和汤姆孙的表述）之间的等效性在低温下可能会被违反。这些效应是热库略微偏离平衡性质情况下量子纠缠的结果，并且当特征量子时间尺度(ћ/2πk_BT)大于或可比于系统的其他时间尺度时变得重要。它们表明标准热力学和量子力学之间没有普遍的共识。已知的一致性仅由于弱耦合极限而出现，而这与低温无关。文中讨论了用于测试这些效应的实验装置。
