In this paper, we consider the particle on curved graphene space-time. In that case, we calculate the geometric form of potential which is known as Gaussian function. Here, we introduce the metric background which completely corresponds to curved graphene space-times. This metric leads us to obtain the geometry potential and we make the Laplace Beltrami equation in the mentioned metric background. We also rearrange such relation in terms of the second-order equation. By using the known polynomial, we solve the particle equation of motion in graphene background. In that case, we arrive the energy spectrum which has three terms. We take advantage from energy spectrum and investigate the thermal properties of system. The additional terms give us an opportunity to obtain the corrected entropy and free energy. So, we show that the additional term comes from geometry potential. This correction is important for the large scale. Hence, we show that correction term is logarithmic as well as small scale corrections.

中文翻译：

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自适应石墨烯模型全息双背景的大尺度校正和热特性

在本文中，我们考虑弯曲石墨烯时空上的粒子。在这种情况下，我们计算称为高斯函数的势的几何形式。在这里，我们介绍完全对应于弯曲石墨烯时空的度量背景。这个度量引导我们获得几何势，我们在提到的度量背景中建立了拉普拉斯贝尔特拉米方程。我们还根据二阶方程重新排列这种关系。通过使用已知多项式，我们求解了石墨烯背景中的粒子运动方程。在这种情况下，我们得到具有三个项的能谱。我们利用能谱来研究系统的热特性。附加项使我们有机会获得校正后的熵和自由能。所以，我们表明附加项来自几何势。这种修正对大规模很重要。因此，我们表明校正项是对数校正以及小规模校正。