Thermodynamics properties of the dark energy in loop quantum cosmology Kui Xiao ∗ and Jian-Yang Zhu † Department of Physics, Beijing Normal University, Beijing 100875, China (Dated: June 27, 2018) Considering an arbitrary, varying equation of the state parameter, the thermodynamic properties of the dark energy fluid in a semiclassical loop quantum cosmology scenario, which we consider the inverse volume modification, is studied. The equation of the state parameters are corrected as a semiclassical one during considering the effective behavior. Assuming that the apparent horizon has Hawking temperature, the modified entropy-area relation is obtained, we find that this relation is different from the one which is obtained by considering the holonomy correction. Considering the dark energy is a thermal equilibrium fluid, we get the expressions for modified temperature, chemical potential and entropy. The temperature, chemical potential and entropy are well-defined in the semiclassical regions. PACS numbers: 04.60.Pp, 04.60.Kz, 98.80.Qc I. INTRODUCTION arXiv:1001.0306v2 [gr-qc] 28 Jun 2010 More and more evidences [1] show that our universe may be dominated by dark energy (for more recent re- view, please read [2] and references therein.). Although the cosmology constant Λ maybe responsible for the ob- served data, but it is also the possible that there are exiting the dynamical mechanism is at work. There are some candidates to describe those dynamical systems, e.g., a canonical scalar field (quintessence) [3], a scalar field with negative sign of kinetic term (phantom) [4], or the combination of quintessence and phantom in a unified model named quintom [5] and so on [2]. Many authors have analysed the characters of dark energy in the clas- sical cosmology, e.g., the inflation caused by dark energy [6], attractor behavior and solution [7], the relationship between geometry and dark energy [8], the fate of uni- verse with dark energy [9], the singularity of universe with dark energy [10], the thermodynamical properties of dark energy [11] and so on. And some authors stud- ied the quantum properties of it [12]. But we still have very little knowledge about it, and there is not any direct evidence shows whether the dark energy exists. So it is necessary to study the properties of it. The universe can be considered as a thermodynamical system. The thermodynamical properties of the universe was studied by many authors [13, 14]. The apparent horizon and event horizon of universe are different, the first and second law of thermodynamics hold on the ap- parent horizon, but they break down while we consider the event horizon [14]. According to [15], the thermo- dynamical properties of the universe in loop quantum cosmology (LQC) is still held. But the corrected Fired- mann equation will modify the entropy-area relation of apparent horizon in LQC [16]. Due to this subtlety, we will focus our attention on the thermodynamical prop- erties of the apparent horizon first. And then we will derive thermodynamical variables of dark energy fluid in the inner side of the apparent horizon in the semiclassical LQC scenario. Loop quantum cosmology (for more recent review, please read [17, 18]) is a canonical quantization of homo- geneous spactimes based upon techniques used in loop quantum gravity (LQG). Due to the homogeneous and isotropic spacetime, the phase space of LQC is simpler than LQG. The connection is determined by a single pa- rameter c and the triad is determined by p . The variables c and p are canonically conjugate with Poisson bracket { c, p } = 8 πG 3 γ , in which γ is the Barbero-Immirzi pa- rameter. In the LQC scenario, the initial singularity is instead by a bounce. So, thanks for the quantum effect, the universe is initially contracting phase with the min- imal but not zero volume, and then the quantum effect drives it to the expanding phase. If one wants to consider the physical effect near the minimal area, e.g., very small scale factor a < a$_{i}$ = √ γl$_{pl}$ , it is necessary to consider the difference equation [19]. But if we consider the interval a$_{i}$ < a < a$_{⋆}$ = √ γj/ 3 l$_{pl}$ , with the quantum parameter j , we can consider the effective dynamics of LQC. And the classical cosmology can be recovered if a > a$_{⋆}$ . In this paper, we just consider the thermodynamical of the dark energy fluid in the semiclassical regions. In the classical regions, the dark energy thermodynam- ics has been studied by many authors [11]. The thermo- dynamics properties of dark energy fluid described by the equation of state parameter ω$_{cl}$ = P$_{cl}$/ρ$_{cl}$ , in which P$_{cl}$, ρ$_{cl}$ are the pressure and energy density of dark en- ergy fluids respective. For the thermodynamical proper- ties of quintessence filed, with a constant ω$_{cl}$ > − 1, it is nature to use the results of conventional perfect fluid thermodynamics [20]. But, in the case of phantom fields, it is more complex if we consider constant ω$_{cl}$ < − 1. Considering the chemical potential µ = 0, some authors showed that the energy density and entropy are positive, but the temperature is negative [21, 22], and they ar- gued that this negative temperature due to the quantum properties of phantom fields. And some other authors $^{∗}$Electronic address: 87xiaokui@mail.bnu.edu.cn $^{†}$Electronic address: zhujy@bnu.edu.cn