在此篇論文之前, 已經有許多學者指出在金融市場奔盤之前的價格波動與熱物理學中的臨界現象有所類似. 其價格會呈現Power law的形式迅速加速上升, 同時伴隨著log-periodic震盪. 藉由first-order Landau expansion和second-order Landau expansion, 我們使用了50個隨機樣本, 分別從五個不同的指數來驗證其正確性. 結果發現該模型很難運用在高波動的市場, 但是對於中級波動的市場卻有不錯的預測能力, 比方說S&P500與Nikkei 225指數.

Before this paper, many scholars indicated that market price movement before a crash is similar to critical phenomena. It can be described by a power law acceleration of the market price decorated with log-periodic oscillations. By first-order Landau expansion and second-order Landau expansion, we use 50 random samples from each of 5 different indices to test the model. It is hard to adapt Landau expansion to high volatility indices, but fit pretty well for medium volatility indices, such as S&P 500 and Nikkei 225.

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